THECON      '.HON  OF  THE  INFINITE 


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REESE    LIBRARY 


UNIVERSITY   OF   CALIFORNIA. 

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THE 


CONCEPTION  OF  THE  INFINITE, 


AND    THE 


SOLUTION  OP  THE  MATHEMATICAL 
ANTINOMIES: 


A  STUDY  IN  PSYCHOLOGICAL  ANALYSIS. 


BY 

GEORGE   S.  FULLERTON,  A.M.,  B.D., 

ADJUNCT   PROFESSOR   OF   PHILOSOPHY   IN   THE    UNIVERSITY   OF   PENNSYLVANIA. 


UNIVERSITY 


PHILADELPHIA  : 

J.   B.   LIPPINCOTT    COMPANY. 

188  7. 


BZ 


Copyright,  1887,  by  George  S.  Fdllertok. 


43%ix^ 


7- 


■llll'sTLREOTYPERSflNI'PHINTI.Ks'lll:' 


PEEFAOE. 


The  question  treated  in  this  little  volume  is 
one  of  no  small  interest  from  several  quite 
different  points  of  view.  To  one  interested  in 
lucid  and  systematic  thinking,  the  tangle  of 
thought  which  has  always  obtained  in  this 
corner  of  the  philosophic  field  cannot  but  be 
repulsive  and  irritating.  To  be  told  that  of 
two  impossible  things  one  must  be  true;  that 
of  the  same  two  lines  one  may  be  looked  upon 
as,  at  pleasure,  equal  to,  less  than,  or  greater 
than  the  other,  both  remaining  unchanged; 
that  Achilles,  running  rapidly,  can  never  over- 
take the  tortoise,  moving  slowly ;  to  be  told 
all  this  seriously,  by  men  whose  calling  it  is 
to  think  and  to  teach  others  to  think,  is  well 
calculated  to  bring  not  merely  suspicion  but 
contempt  upon  speculative  thought,  and  de- 
servedly.     Who   has  not  puzzled,  on  his   first 

3 


4  PREFACE. 

introduction  to  Logic,  over  some  of  these  an- 
tinomies, and  been  silenced  unconvinced  by  the 
practical  demonstration, — as  by  walking,  in  the 
case  of  the  argument  against  motion, — which 
cuts  the  knot  but  does  not  solve  it,  leaving  in 
the  mind  a  disagreeable  sense  that  the  argu- 
ment must  be  wrong  somewhere,  and  yet  a 
consciousness  that  it  certainly  seems  perfectly 
sound?  When  the  metaphysician  proves  to  us 
that  a  rhinoceros  is  a  mosquito,  his  chain  of 
reasoning  is  rendered  innocuous  by  the  striking 
incongruity  of  the  conclusion ;  but  if  we  ob- 
serve no  flaw  in  his  reasoning,  we  cannot  help 
recognizing  the  perplexing  truth  that  it  is  the 
experienced  fact  alone  which  has  prevented  as- 
sent, and  that  a  precisely  similar  argument,  the 
conclusion  of  which  cannot  be  similarly  tested, 
may  yet  induce  assent,  though  equally  errone- 
ous. If  we  have  no  better  reason  for  rejecting 
an  argument,  what  can  be  our  criterion  when 
we  leave  the  sphere  of  the  immediately  palpa- 
ble? He  who  has  convinced  himself  that  the 
minute  hand  of  a  clock  cannot  overtake  the 
hour  hand,  will  be  enlightened  when  the  clock 
strikes  at  noon ;  but  he  who  has  followed  Mr. 
Spencer  into  his  discussions  regarding  our  no- 


PREFACE.  5 

tions  of  infinite  space  or  time,  will  be  filled 
with  inward  dismay  if  he  hang  his  hope  upon 
any  such  practical  expedient.  Civil  history  can- 
not be  studied  in  the  laboratory,  nor  erroneous 
ideas  as  to  infinite  space  rectified  with  the  aid 
of  the  foot-rule.  In  this  sphere,  too,  the  ques- 
tion of  a  careful  and  thorough  analysis  of  our 
conception  of  the  infinite  is  of  more  than  a 
merely  intellectual  interest,  and  any  erroneous 
conception  which  can  blossom  out  into  such  a 
development  as  the  "  Philosophy  of  the  Condi- 
tioned," with  its  implications,  has  a  religious 
significance  which  cannot  be  overlooked.  The 
analysis  of  this  single  conception  is,  moreover, 
of  importance  as  throwing  light  upon  the  pro- 
cedure of  thought  in  general,  and  will  to  many 
be  of  more  interest  in  this  connection  than  for 
its  own  sake.  I  have  endeavored  to  write  with 
extreme  clearness  and  simplicity,  and  to  avoid, 
as  much  as  possible,  all  issues  not  directly  con- 
nected with  the  immediate  subject;  and  whether 
my  discussion  meet  with  assent  or  dissent,  I  do 
not  think  it  will  be  charged  with  the  obscurity 
characteristic  of  discussions  upon  this  much- 
mooted  topic. 

Portions  of  the  book  are  reprinted,  with  ad- 


6  PREFACE. 

ditions  and  alterations,  from  the  American  Jour- 
nal of  Speculative  Philosophy  and  from  the  Brit- 
ish  periodical    Mind,  in   which   they  originally 
appeared. 
University  of   Pennsylvania,  December,  1886. 


CONTENTS. 


CHAPTEK    I. 

PAGE 

Introductory  9 


CHAPTEK    II. 
The  Conception  not  Quantitative  .        .        .        .20 

CHAPTEK    III. 
The  Antinomies  of  Hamilton 34 

CHAPTEK    IV. 
Kant,  Mill,  and  Clifford 53 

CHAPTEK    V. 

The  Conceivable  and  the  Existent        .        .        .     77 

CHAPTEK    VI. 

The  Conceivability  of  the  Infinite       .         .         .90 


THE 

CONCEPTION  OF  THE  INFINITE, 


AND    THE 


SOLUTION    OF    THE    MATHEMATICAL 
ANTINOMIES: 

A  STUDY  IN  PSYCHOLOGICAL  ANALYSIS. 


CHAPTER    I. 

INTRODUCTORY. 

The  doctrine  that  there  may  be  an  infinity  of 
worlds,  thought  Plutarch,*  is  to  be  repudiated. 
Providence  could  not  possibly  take  charge  of  so 
many;  "troublesome  and  boundless  infinity" 
could  be  grasped  by  no  consciousness. 

Plutarch's  decision  as  to  the  unknowability 
of  the  infinite  the  student  of  the  history  of 
thought  will  find  reiterated  by  thinkers  who 
agree  upon  little  else  than  the  one  point,  that 


*  "  De  defectu  oraculorum,"  c.  24. 


10       THE   CONCEPTION  OF  THE  INFINITE. 

when  Ave  leave  the  finite  and  talk  of  the  infinite 
we  are  playing  with  a  word, — deceiving  our- 
selves into  believing  that  we  can  know  what  is 
in  its  very  nature  inconceivable. 

The  notion  of  the  infinite,  it  is  said,*  is  a 
negative  one :  all  our  experience  of  objects 
being  of  them  as  finite,  we  can  think  to  our- 
selves the  negation  of  this  condition,  and  thus 
form  a  negative  conception, — i.e.,  not  an  affir- 
mation of  a  quality  or  attribute,  but  a  simple 
denial  of  a  quality  known.  But  to  know 
an  object  —  for  example,  space  —  as  infinite, 
this  is  beyond  our  power.  The  human  mind 
is  finite;  all  which  can  become  an  object  of 
consciousness  is  finite;  and  "troublesome  and 
boundless  infinity"  cannot  be  an  object  of 
thought. 

"  Whereas,  for  this  very  reason,"  said  Cud- 
worth,f  "  because  more  could  be  added  to  the 
magnitude  of  the  corporeal  world  infinitely,  or 
without  end,  therefore  it  is  impossible,  that  it 
should  ever  be  positively  and  actually  infinite ; 


*  See  Hamilton's  Discussions,  u  Philosophy  of  the  Uncon- 
ditioned." 

f  "Intellectual  System,"'  chap,  v.,  Andover,  1838,  vol.  ii. 
p.  45. 


INTRODUCTORY.  \\ 

that  is  such  as  to  which  nothing  more  can  pos- 
sibly be  added." 

A  favorite  position :  all  attempts  at  cognizing 
the  infinite  must  result  in  the  indefinite;  for 
the  infinite  can  only  be  known  by  the  progres- 
sive addition  of  finites,  an  addition  which  can 
be  completed  only  in  an  infinite  time ;  in  other 
words,  can  never  be  completed.  Each  stage  in 
the  addition  gives  but  the  indefinite,  and  the 
last  stage,  the  infinite,  is,  by  the  terms  of  the 
problem,  unattainable. 

How,  indeed,  it  is  asked,  could  a  finite  mind 
know  the  infinite?  "Adequately  to  know  what 
is  infinite  is  to  have  infinite  knowledge."*  We 
cannot,  surely,  lay  claim  to  that. 

The  famous  antinomies  of  the  philosopher  of 
Konigsberg  have,  more  than  anything  else, 
stirred  up  the  minds  of  men  to  a  consideration 
of  the  problem.  The  indefinite,  said  Kant,  we 
may  know  as  a  whole,  not  by  passing  succes- 
sively over  each  of  its  parts,  but  immediately 
as  a  unit,  by  means  of  its  limits ;  but  in  the  case 
of  the  infinite,  since  there  are  no  limits,  we 
must  arrive   at  our  cognition   by  a  successive 

*  "  The  Battle  of  the  Two  Philosophies,"  p.  24. 


12       THE   CONCEPTION  OF  THE  INFINITE. 

addition  of  parts,  which  addition  must  neces- 
sarily be  itself  endless,  and  therefore  the  attempt 
thus  to  know  the  infinite  futile.  Although  we 
are  unable  to  conceive  of  an  absolute  commence- 
ment to  time  or  of  an  absolute  limit  to  space, 
we  are  not  on  that  account  justified  in  assuming* 
either  to  be  infinite,  as  this  we  can  only  know 
when  we  have  passed  successively  over  all  spaces 
and  all  times.  The  whole  question  is  one  the 
decision  of  which  is  beyond  the  scope  of  human 
reason;  both  alternatives  are  equally  inconceiv- 
able. 

The  doctrine  of  the  inconceivability  of  the 
infinite  is  much  dwelt  upon  by  Sir  William 
Hamilton  as  one  phase  of  his  cherished  theory, 
the  Philosophy  of  the  Conditioned.  He  argues 
that  space,  for  example,  cannot  be  cognized  as 
either  infinitely  extended  and  infinitely  divisible, 
or,  on  the  other  hand,  as  absolutely  limited, 
whether  as  a  maximum  or  minimum,  just  as  Kant 
argued.  Mr.  Mansel  has  accepted  and  de- 
fended* the  positions  taken  by  Hamilton;  while 
the   same   arguments,   which  are   used    by    Sir 


*  See   Mr.   Mansel's    "Philosophy   of    the   Conditioned.1 
London:  1866. 


INTRODUCTORY.  13 

William  Hamilton  and  his  follower,  Mr.  Man- 
sel,  to  elevate  faith  at  the  expense  of  reason  or 
science,  are  to  be  found  upon  the  pages  of 
the  "  First  Principles  of  Philosophy,"  where 
they  are  used  by  Mr.  Spencer  in  support  of  a 
philosophy  widely  different  from  the  Hamil- 
tonian. 

It  would  not  be  difficult  to  multiply  testimo- 
nies to  the  inconceivability  of  the  infinite,  for 
the  misconception  which  we  find  in  Plutarch 
appears  and  reappears  in  divers  forms  in  dif- 
ferent ages  and  climes,  much  as  the  Wandering 
Jew  may  be  supposed  to  have  presented  him- 
self; and,  to  carry  out  the  simile,  the  well-worn 
dress  in  which  the  doctrine  usually  comes  to  the 
surface — the  statement  that  we  can  know  the 
infinite  only  by  an  endless  addition  of  finites — 
may  be  not  inaptly  compared  to  the  threadbare 
garment  on  his  back. 

It  would  seem  surprising  that  there  could  be 
so  universal  a  misconception  with  regard  to  the 
nature  of  a  conception  actually  present  at  some 
time  or  other  in  the  mind  probably  of  every 
man,  and  in  the  case  of  many  not  unfrequently 
present;  a  conception  sufficiently  familiar  and 
important  to  be  recognized  and  marked  by  its 


14       THE   CONCEPTION   OF  THE  INFINITE. 

appropriate  name.  But  a  very  little  knowledge 
of  psychological  processes  will  make  one  cogni- 
zant of  the  fact  that  the  interpretation  of  con- 
sciousness is  by  no  means  the  easy  and  simple 
task  that  by  a  novice  it  might  be  supposed 
to  be. 

There  are  but  few  who  have  an  analytical 
knowledge  of  even  the  most  common  of  their 
mental  operations ;  and  the  average  man  is  very 
literally  incapable  of  telling  what  may  at  any 
moment  be  passing  in  his  mind.  It  is  a  matter 
of  surprise,  for  instance,  to  one  unaccustomed  to 
psychological  analysis,  to  learn  that  distance  is 
not  directly  perceived  by  vision,  but  that  judg- 
ments of  distance  are  the  result  of  a  rapid  pro- 
cess of  reasoning,  and  imply  a  generalization 
from  past  experience.  The  judgment  seems  in- 
stantaneous and  intuitive.  Those  elements  which 
are  in  fact  visual  and  actually  present  are  not 
distinguished  from  those  present  only  by  sug- 
gestion ;  that  is,  present  in  the  imagination. 
The  mental  state  is  grasped  as  a  unit,  and  for 
practical  purposes  no  analysis  into  its  elements 
is  necessary.  So  it  is  with  the  concept,  or  gen- 
eral notion.  That  men  form  general  notions,  or 
at  least  represent  in  mind  objects  and  their  re- 


INTR  OD  VCTOR  Y.  15 

lations  in  some  way  different  from  that  in  which 
simple  intuitions  are  singly  represented,  all  are 
ready  to  acknowledge.  But  that  the  psychical 
elements  concerned  in  this  act  are  not  clearly 
apprehended  is  evident  from  the  common  war- 
fare of  Realism,  Nominalism,  and  Conceptual- 
ism.  So  is  it,  again,  in  the  case  of  memory. 
The  act  is  usually  described  as  if  there  were 
present  in  the  mind  the  two  elements  of  a  pres- 
ent mental  representation  and  an  intuition,  or 
presentation,  its  prototype,  to  which  it  is  re- 
ferred ;  while  the  picture  before  the  mind  is  but 
one,  as  an  examination  of  consciousness  during 
the  act  will  very  readily  show.  And  the  same 
truth  is  illustrated  in  innumerable  ways  by 
the  debates  and  disputes  of  philosophers,  past 
and  present,  as  to  mind  and  the  faculties  in 
which  it  is  manifested.  As  a  typical  instance 
may  be  given  the  analysis  of  consciousness  left 
us  by  Mr.  John  Stuart  Mill  as  compared 
with  that  presented  in  Sir  William  Hamilton's 
Lectures.  Was  not  each  describing  what  was 
present  in  his  own  mind?  Whence  the  discre- 
pancy ? 

Moreover,   a  clear  apprehension  of  the  con- 
stituent  elements   of  a  mental   state   does   not 


1(3        THE   CONCEPTION  OF  THE  INFINITE. 

always  seem  to  be  necessary,  from  a  practical 
point  of  view,  to  enable  one  to  use  that  state 
as  a  unit,  with  substantial  accuracy.  The  Nom- 
inalist, the  Realist,  and  the  Conceptualist  all 
speak  the  same  language,  refer  to  the  same 
objects,  and,  in  the  use  of  the  complex  mental 
phenomenon  which  they  so  variously  analyze, 
are  at  one.  A  seaman,  practised  in  the  judg- 
ment of  distances  by  a  long  training  in  his  vo- 
cation, may  be  much  more  accurate  in  his  judg- 
ments of  distance  than  the  accomplished  author 
of  the  "New  Theory  of  Vision"  himself,  though 
he  may  be  quite  ignorant  of  the  mental  process 
by  which  he  arrives  at  his  conclusions.  The 
process  itself  is  not  directly  affected  by  an  ana- 
lytic consciousness  of  the  steps  of  the  process, 
nor  is  its  result.  The  mental  state  is  in  most 
cases  recognized  only  as  a  unit,  since  it  is  as  a 
unit  that  it  is  useful  to  the  individual ;  and  the 
name  by  which  it  is  known  expresses  the  gen- 
eral impression  conveyed  to  the  mind  when  the 
state  is  called  up. 

JSToW,  it  is  manifest,  that  when  one  begins  to 
analyze  this  vague  and  indefinitely  grasped 
total,  and  to  separate  it  into  its  elements,  it  is 
quite  possible  for  him  to  confound  some  of  the 


/3erir<u, 


INTROD  UCTOR  Y.  17 

elements  with  elements  somewhat  similar,  or 
to  imagine  the  presence  of  elements  closely 
connected  by  the  laws  of  association  with  ele- 
ments really  present ;  in  short,  to  find  what  is 
not  there,  and  what,  when  in  practice  he  uses 
the  word  indicating  the  state  as  a  unit,  he  never 
means  to  express  by  it.  And,  in  view  of  this 
fact,  we  may  see  how  it  is  possible  for  the  curi- 
ous error  regarding  the  inconceivability  of  the 
infinite,  which  has  been  adverted  to,  to  have 
arisen,  and  to  have  held  its  place  in  the  writings 
of  philosophers.  The  word  has  always  been 
used,  and  the  ideas  for  which  it  stands  have 
often  been  in  men's  minds;  but  in  attempting 
to  explicate  the  conception,  we  find  that  almost 
all  place  among  the  qualities  it  connotes  a 
notion  drawn  from  finites,  and  which  is  contra- 
dictory to  the  essential  character  of  the  concep- 
tion. Few  men  have  talked  more  about  the 
infinite  than  Sir  William  Hamilton ;  and  it  is 
probable  that  every  time  he  used  the  word  it 
called  up  very  much  the  same  mental  state  in 
his  mind  as  that  which  arose  in  the  mind  of 
Mr.  John  Stuart  Mill  when  he  read  Sir  "Wil- 
liam's Lectures ;  but  the  conception  thus  called 
up  certainly  did   not   contain   the  warring  ele- 

2* 


18        THE   CONCEPTION   OF  THE  INFINITE. 

ments  which  Sir  "William  finds  in  it  when  he 
undertakes  to  prove  it  unthinkable.  And  when 
Mr.  Mill  criticises  Sir  William  Hamilton,  and 
declares  the  infinite  not  inconceivable,  he  prob- 
ably meant  by  the  infinite  just  what  Hamilton 
did  ;  and  yet  when  he  tries  to  prove  its  con- 
ceivability,  he  finds  in  it  what  was  not  really 
contained  in  the  mind  of  either  when  the  word 
was  used. 

The  problem  is  simply  one  of  psychological 
analysis, — a  question  of  what  may  be  the  true 
content  of  a  complex  mental  state ;  and  the 
fact  that  such  an  analysis  may  be  incorrectly 
made  need  not  surprise  us,  since  erroneous  an- 
alyses are  only  too  common.  What  may  well 
create  surprise,  however,  is  that  the  difficulties 
into  which  this  erroneous  analysis  has  led  those 
guilty  of  it,  the  antinomies  of  which  it  is  the 
source,  have  not  called  attention  to  the  error, 
and  that  a  rigorous  analysis  has  not  been  em- 
ployed to  eliminate  it.  I  will  consider  in  the 
remaining  chapters  of  this  little  book  the 
errors  which  have  arisen  from  a  mistaken  no- 
tion of  the  content  of  our  conception  of  the 
infinite,  and  show  that  they  all  arise  from  a 
foreign  and  contradictory  element  inadvertently 


INTRODUCTORY.  19 

admitted  as  part  of  the  conception,  and  that, 
this  element  being  eliminated,  the  conception 
is  in  no  respect  inconceivable,  nor  does  it  pre- 
sent any  difficulties  not  presented  by  any  other 
concept  or  general  notion. 


CHAPTER    II. 

THE   CONCEPTION   NOT   QUANTITATIVE. 

We  will  suppose  two  parallel  straight  lines, 
A  and  B,  unlimited  in  extent,  and  intersected 
by  perpendiculars,  ab,  a'b',  a"b" ,  etc.,  drawn  at 
equal  distances  from  each  other. 


b         b'        b" 

It  is  evident  that  each  division  upon  A  is  equal 

to  its  corresponding  division  upon  B,  and  the 

sum  of   any  number  of  divisions  upon  A  will 

equal  the    sum   of   a   similar   number  upon  B. 

Since,  therefore,  each  division  upon  the  one  line 

has  its   corresponding  division  upon  the  other, 

will  not  the  equation  hold  good  when  all   the 

divisions  are  considered  ?     That  is,  will  not  the 

sum  of  all  the  divisions  upon  A  be  equal  to  the 

sum  of  all  the  divisions  upon  B  ?     And  will  not 

the  sum  of  all  the  divisions  on  both  lines  be 

equal  to  twice  the  sum  of  all  the  divisions  on 

either?     Must  we  not  here  regard  one  infinite 

as  greater  than  another  ? 
20 


THE   CONCEPTION  NOT  QUANTITATIVE.     21 

Much  depends  upon  the  answer  to  this  ques- 
tion, as  it  will  reveal  very  clearly  the  content 
which  one  attributes  to  his  conception  of  the 
infinite.  To  the  giving  of  the  wrong  answer 
may  be  traced  that  misconception  of  the  true 
nature  of  infinity  which  has  been  such  a  fruitful 
source  of  supposed  antinomies.  Broadly  stated, 
the  question  is,  Can  infinites  be  regarded  as 
comparable  with  each  other,  as  greater  or  less 
than,  or  equal  to,  each  other  ?  Let  us  consider 
the  case  of  the  parallel  lines. 

It  is  true  that  we  must  consider  each  divis- 
ion on  the  one  line  equal  to  each  division  on 
the  other ;  and  taking  any  number  of  divisions 
on  the  one,  and  adding  them  to  an  equal  num- 
ber of  divisions  on  the  other,  we  obtain  a  sum 
equal  to  twice  the  number  of  given  divisions  on 
either.  But  when  we  say  "  all  the  divisions  on 
the  one  are  equal  to  all  the  divisions  on  the 
other,"  we  speak  of  the  lines  as  quantitative 
wholes,  and  introduce  an  error  with  the  word 
all. 

To  conceive  of  a  thing  as  a  whole,  we  must 
assign  to  it  limits.  In  saying  "  the  whole"  of 
any  object,  we  refer  to  those  limits  beyond 
which   there   is   none    of  that   object.      In   re- 


22       THE   CONCEPTION  OF   THE  INFINITE. 

garding  any  object  as  a  quantitative  whole,  we 
necessarily  think  it  as  finite. 

When  we  compare  one  line  with  another, 
and  declare  its  extent  greater  or  less  than  that 
of  the  other,  we  mean  that,  when  the  one  is 
applied  to  the  other,  its  limits  extend  be- 
yond or  fall  within  the  limits  of  the  other. 
In  other  words,  we  give  the  difference  between 
the  distances  included  within  their  respec- 
tive limits.  Measuring  is  merely  giving  the 
distance  between  limits.  When  two  lines  are 
infinite,  we  have  no  point  to  measure  from, 
and  no  point  to  measure  to,  and  no  measure- 
ment—  therefore  no  comparison — is  possible. 
It  is  a  'palpable  contradiction  to  compare  (i.e., 
give  relations  of  measurement  between  the  re- 
spective limits  of)  two  infinites  (i.e.,  things 
which  cannot  be  measured,  as  having  no 
limits). 

The  terms  longer,  shorter,  and  equal,  can,  there- 
fore, have  no  meaning  as  applied  to  infinite  lines. 
They  can  be  used  only  in  speaking  of  the  finite. 
We  cannot,  then,  say  that  one  infinite  is  greater 
or  less  than  another,  and  just  as  little  can  we 
say  that  all  infinites  are  equal ;  for  any  such 
proposition,  however  possible  in  words,  is   im- 


THE   CONCEPTION  NOT  QUANTITATIVE.      23 

possible  in  thought,  and  is  an  attempt  to  join 
contradictory  notions. 

In  such  cases  as  the  above,  where  the  lines 
are  nowhere  limited,  the  impossibility  of  an  in- 
crease in  length  may  be  clearly  seen.  A  line 
can  only  be  lengthened  by  adding  to  it  at  its 
extremities,  and  it  is  impossible  that  a  line  with- 
out ends  should  be  added  to.  If  one  holds  that 
the  sum  of  two  such  lines  is  greater  than  either 
line  separately,  he  simply  states  that  that  may 
be  increased,  the  very  conception  of  which  pre- 
cludes the  possibility  of  its  increase. 

There  are  cases,  however,  in  which  the  error 
of  a  wrong  conclusion  is  not  so  immediately  pal- 
pable as  in  the  case  just  stated;  for  example,  the 
case  of  a  line  limited  at  but  one  point.  Suppose 
the  line  AB  limited  only  at  the  point  A. 


Continue  the  line  to  C.  If  now  the  line  be  di- 
vided by  points  placed  at  equal  distances  from 
each  other,  into  equal  divisions,  AC  containing 
three  such  divisions,  will  not  the  whole  line  CB 
be  greater  by  three  divisions  than  the  whole 
line  AB? 

AB  is   limited  at  A;     consequently  there  is 


24       THE   CONCEPTION   OF  THE  INFINITE. 

nothing  to  prevent  our  adding  to  it  at  its  one 
extremity.  Does  it  not  seem  natural  to  assume 
that  in  thus  adding  we  increase  the  sum  total  of 
the  line  ?  We  have  gone  through  the  same 
process  as  that  by  which  we  increase  finite 
lines. 

When  we  recollect,  however,  that  the  line 
AB  is  limited  only  at  one  point,  and  is  not, 
therefore,  as  a  line,  defined  (for  two  points  are 
necessary  to  define  a  line),  the  impossibilit}'  of 
regarding  it  as  a  quantitative  whole  is  evident, 
and  the  impossibility  of  increasing  or  diminish- 
ing its  length,  as  a  wThole,  necessarily  follows. 
All  of  CB  is  not  greater  than  all  of  AB,  be- 
cause the  word  all  (in  its  quantitative  sense*) 
cannot  be  applied  to  either. 

Suppose  we  attempt  a  comparison  of  the  two 
lines.  Let  CB  be  superposed  upon  AB  in 
such  a  manner  that  C  will  fall  upon  A.  The 
two  lines  will  then  have  the  one  limit  in  com- 
mon ;  but  one  limit  does  not  furnish  data  for 
lineal  comparison,  and  no  judgment  can  be 
formed  as  to  the  comparative  length  of  the 
two  lines.     Where  the  one  line  is  regarded  as 

*  This  distinction  will  be  noticed  later. 


THE   CONCEPTION  NOT   QUANTITATIVE.      25 

greater  than  the  other,  from  the  fact  that  three 
of  its  divisions  project  heyond  the  limit  of  the 
other,  the  measurement  begins  with  an  im- 
agined point  infinity,  which  is  regarded  as  a 
common  limit  of  the  other  extremities  of  the 
two  lines,  and  concludes  with  the  limits  at  A 
and  C. 

The  error  of  such  an  attempt  at  measurement 
is  clearly  revealed  by  beginning  to  measure  at 
A  and  C.  It  may  be  here  justly  remarked 
that  we  have  before  us  a  concrete  instance  of 
the  truth  of  the  old  adage,  that  is  a  poor  (meas- 
uring) rule  that  will  not  work  both  ways. 
The  illusion  disappears  when  we  begin  to 
measure  at  the  given  and  only  limits. 

Now,  drawing  the  necessary  inference  from 
the  foregoing,  we  may  answer  the  question 
whether  a  line  altogether  without  limits  is  not 
greater  than  a  line  limited  at  but  one  point,  by 
saying  that  the  very  nature  of  the  conceptions 
precludes  the  possibility  of  the  words  greater  or 
less  being  applied  to  either ;  that  neither  of  the 
lines  can  be  regarded  as  a  quantitative  whole ; 
and  that,  consequently,  the  question  is  a  mean- 
ingless one. 

When  we  turn  our  attention  to  the  considera- 


26        THE   CONCEPTION  OF  THE  INFINITE. 

tion  of  surfaces,  we  meet  with  similar  misappre- 
hensions, and  arising  from  the  same  cause.  It 
is  asserted,  for  example,  that  if  we  suppose  three 
parallel  straight  lines  infinite  in  extent,  one  of 
which,  A,  is  separated  by  a  distance  of  two 
metres  from  the  middle  line  B,  while  the  other, 
C,  is  distant  from  it  but  one  metre, 


we  must  conclude  that  the  surface  included  be- 
tween A  and  B  is  double  the  surface  included 
between  B  and  C. 

Let  the  lines  be  intersected  by  perpendiculars 
one  metre  apart.  Do  we  not  find  that  each 
metre  in  the  surface  contained  between  B  and 
C  has  corresponding  to  it  two  metres  in  the 
other  surface?  And  is  not  this  proportion  the 
same  for  twenty  or  for  two  hundred  as  for  one, 
and  quite  independent  of  number?  If,  then,  the 
proportion  will  hold  good  for  any  number,  why 
will  it  not  hold  good  when  all  the  divisions  are 
considered  ? 

Or  if  we  consider  the  angle  formed  by  the 


THE   CONCEPTION  NOT  QUANTITATIVE.     27 

intersection  of  two  infinite  straight  lines,  must 
we  not  conclude  that  increasing  the  anscle  will 
increase  the  area  of  the  surface  included  be- 
tween the  lines  forming  the  angle  ?  And  that 
diminishing  the  angle  will  diminish  the  area  of 
the  included  surface  ?  Let  A  and  B  be  two 
infinite  straight  lines  intersecting  at  C ; 


and  let  A'C  be  an  infinite  straight  line,  making 
a  smaller  angle  with  CB  than  is  made  by  AC. 
Must  we  not  affirm  that  the  surface  ACB  is 
greater  than  the  surface  A'CB,  and  that  it  is 
equal  to  the  sum  of  A'CB  and  ACA'?  The 
answer  to  be  given  in  these  cases  is  evidently 
the  answer  which  has  already  been  given.  The 
quantitative  relations  of  equality  and  inequality 
certainly  hold  good  for  all  quantities;  and 
taking  the  case  of  the  surfaces  included  be- 
tween the  parallel  lines,  we  must  admit  that  any 
number  of  the  divisions  between  A  and  B  will 


28        THE   CONCEPTION  OF  THE  INFINITE. 

have  double  the  area  of  a  similar  number  of  the 
divisions  between  B  and  C ;  but  when  we  speak 
of  all  the  divisions,  we  do  not  refer  to  any 
number;  we  do  not  express  by  the  word  a 
quantity;  and  where  the  notion  of  quantity  is 
wanting,  manifestly,  quantitative  relations  will 
not  hold.  So  in  the  case  of  the  increased  or 
diminished  angle. 

It  is  unnecessary  to  multiply  instances,  as  the 
principle  is  in  all  cases  the  same.  In  general, 
wherever  the  limit  is  removed  in  any  one  direc- 
tion, whether  in  the  case  of  lines,  of  surfaces,  or 
of  solids,  the  object  can  no  longer  be  regarded 
as  a  quantitative  whole,  and  is  not  to  be  con- 
sidered finite. 

It  remains  to  consider  a  class  of  cases  of  an 
apparently  different  nature.  It  is  argued,  for 
example,  that  an  infinite  series  of  dollars  will 
exceed  in  value  an  infinite  series  of  cents ;  that, 
where  the  unit  differs,  the  difference  will  extend 
to  the  series  in  its  totality.  It  is  easy  to  show 
the  error  of  such  a  position  by  showing  "what 
the  assertion  necessarily  involves. 

Suppose  that,  instead  of  counting  one  cent  in 
the  one  series  to  each  dollar  in  the  other,  we 
vary  our  mode  of  procedure   by  counting  one 


THE   CONCEPTION  NOT  QUANTITATIVE.     29 

hundred  cents  in  the  one  to  each  dollar  in  the 
other.  It  is  true  that  the  one  series  is  exhausted 
one  hundred  times  as  rapidly  as  the  other;  but 
since  they  are  both  infinite  (will  never  end),  we 
may  continue  this  forever  (to  infinity),  and  the 
two  series  will  have  equal  values.  Or  we  may 
count  two  hundred  cents  in  the  one  to  each 
dollar  in  the  other,  or  three  hundred,  or  four 
hundred;  so  that  the  same  series  will  be  equal 
to,  or  twice,  thrice,  or  four  times  as  great  as 
another;  its  value  depending  merely  on  the 
mode  of  reckoning. 

If  it  is  just  to  conclude  that  an  infinite  series 
of  dollars  is  one  hundred  times  as  great  in  value 
as  an  infinite  series  of  cents,  we  must  also  accept 
the  conclusions  arrived  at  by  the  other  modes 
of  reckoning;  all  are  based  on  the  same  prin- 
ciple. Our  only  escape  from  these  warring  con- 
clusions is  to  declare  the  principle  underlying 
all  the  modes  of  reckoning  an  erroneous  one. 
The  error  lies  in  regarding  these  infinite  series 
as  in  any  way  capable  of  being  compared  with 
each  other;  in  looking  upon  them  as  quanti- 
tative wholes. 

It  is  clear,  therefore,  that  the  true  conception 
of  the  infinite  is  not  quantitative  but  qualitative,  a 

3* 


30       THE   CONCEPTION   OF   THE  INFINITE. 

fact  which  has  been  very  generally  overlooked, 
and  with  disastrous  consequences  much  the  same 
in  all  cases. 

I  have  said  that  the  word  "  whole,"  as  pre- 
dicating totality,  cannot  be  applied  to  infinites, 
as  naturally  follows  from  the  qualitative  nature 
of  our  conception  of  the  infinite;  but  the  use 
of  the  word  is  really  unavoidable,  and,  when 
it  is  used,  it  should  be  borne  in  mind  that,  ap- 
plied to  infinites,  the  word  has  a  certain  quali- 
tative sense  quite  different  from  that  in  which 
it  is  applied  to  finites.  If  I  were  to  speak  of 
"  all  (possible)  men"  as  distinguished  from 
"  all  (possible)  angels,"  I  should  in  no  respect 
limit  the  infinite  possible  number  of  either 
(the  phrase  "infinite  number,"  though,  strictly 
considered,  incorrect,  may  be  understood  in  a 
qualitative  sense  as  expressing  unlimited  units), 
and  my  conceptions  would  be  not  quanti- 
tative but  qualitative;  for  the  mind  would  be 
occupied,  not  with  the  number  of  objects,  but 
with  certain  conditions  which  any  object  must 
satisfy  to  fall  within  the  one  class  or  the 
other. 

If,  on  the  other  hand,  I  speak  of  "  all  (actual) 
men,"  I  may  regard  them  as  a  definite  known 


THE   CONCEPTION  NOT   QUANTITATIVE.     31 

or  unknown  number,  and  my  conception  may 
be  quantitative. 

Similarly,  if,  in  speaking  of  an  infinite  line,  I 
should  say,  "  the  line  AB  is  in  all  its  parts  a 
straight  line,"  I  could  only  mean  that  one  of 
the  general  conditions  of  the  line  is  straight- 
ness,  and  that,  whatever  part  of  it  may  be 
thought  of,  it  must  agree  with  this  condition. 
The  "  all"  is  in  such  a  case  equivalent  to  "any," 
not  to  "every,"  and  the  two  meanings  may 
easily  be  confounded.  Indeed,  there  is  some- 
thing misleading  in  the  very  expression,  "  an 
infinite  line,"  for  the  unreflective  mind  is  apt 
to  regard  the  object  to  which  it  is  applied  as  a 
unit,  a  whole ;  it  is  very  necessary  in  using  the 
phrase  to  keep  in  mind  its  true  meaning. 

It  may  be  objected  to  what  I  have  here 
brought  forward  that  any  theory  which  denies 
that  we  have  knowledge  of  the  infinite  as  a 
whole  may  justly  be  called  agnostic.  If  we  do 
not  know  the  infinite  as  a  whole,  do  we  not 
know  only  its  parts,  which  are  finite  ?  And 
have  we  any  true  knowledge  of  the  infinite  at 
all? 

I  answer,  the  conception  of  a  part  is,  as  well 
as  the  conception  of  a  whole,  quantitative,  and 


32       THE    CONCEPTION  OF  THE  INFINITE. 

an  object  recognized  as  part  of  a  greater  object 
is  thereby  necessarily  recognized  as  finite.  But 
if  the  object  before  the  mind  is  not  quantita- 
tively regarded  at  all,  either  as  whole  or  part, 
our  conception  may  be  of  the  infinite.  The 
plausibility  of  the  objection  arises  from  its  con- 
founding two  very  different  things,  the  distinc- 
tion between  which  will  be  more  clearly  drawn 
in  the  last  chapter  of  this  monograph. 

But  as  a  preliminary  answer  to  the  objection, 
I  may  say  that  the  assertion  that  we  do  not 
know  the  infinite  as  a  whole  is  by  no  means 
equivalent  to  the  assertion  that  we  do  not  know 
the  infinite.  We  do  not  know  the  moon  as 
square,  but  that  would  scarcely  prove  that  we 
have  no  knowledge  of  the  moon,  since  the 
notion  of  squareness  forms  no  part  of  a  true 
knowledge  of  that  object.  Just  as  little  is  the 
quantitative  conception  of  totality  necessary  to 
a  knowledge  of  the  infinite. 

It  is  not  agnosticism  to  declare  the  mind 
unable  to  think  that  which  is  in  its  nature  self- 
contradictory, — to  define  an  object  as  infinite 
and  then  think  it  as  limited ;  while,  on  the 
other  hand,  any  theory  which  maintains  that 
we  may  know  as  a  whole  that  which,  in  its  very 


THE   CONCEPTION  NOT   QUANTITATIVE.     33 

conception,  precludes  the  possibility  of  its  being 
so  considered,  may  be  accused  of  the  direst 
agnosticism,  as  discrediting  a  fundamental  law 
of  thought,  the  law  of  non-contradiction.  The 
theory  attacked  may  as  a  last  resource  avail 
itself  of  the  old  argumentum  ad  hominem,  and  re- 
mark in  pointed  terms  that  the  kettle  is  not  as 
black  as  some  other  vessels  in  the  speculative 
kitchen. 


CHAPTER    III. 

THE  ANTINOMIES   OF   HAMILTON. 

The  evils  resulting  from  overlooking  the  fact 
that  the  conception  of  the  infinite  is  qualitative 
are  evident  when  we  examine  some  of  the  rea- 
sonings based  upon  the  supposition  that  the 
conception  contains  a  quantitative  element. ' 

And  first  I  will  examine,  so  far  as  it  touches 
the  point  in  question,  that  agnostic  theory  de- 
veloped by  Sir  William  Hamilton  under  the 
name  of  the  Philosophy  of  the  Conditioned, 
the  fundamental  principle  of  which,  is  that  "  all 
that  is  conceivable  in  thought  lies  between 
two  extremes,  which,  as  contradictory  of  each 
other,  cannot  both  be  true,  but  of  which,  as 
mutual  contradictories,  one  must."*  The  rea- 
soning in 'which  Sir  William  applies  this  prin- 
ciple to  our  knowledge  of  space  is  worthy  of 
attention.  " .  .  .  We  conceive  space,"  he  says, 
— "  we  cannot  but  conceive  space.  I  admit, 
therefore,  that  Space,  indefinitely,  is  a  positive 

*  »  Metaphysics,"  New  York,  1880,  p.  527. 
34 


THE  ANTINOMIES   OF  HAMILTON.  35 

and  necessary  form  of  thought.  But  when 
philosophers  convert  the  fact,  that  we  cannot 
but  think  space,  or,  to  express  it  differently, 
that  we  are  unable  to  imagine  anything  out  of 
space, — when  philosophers,  I  say,  convert  this 
fact  with  the  assertion  that  we  have  a  notion, 
a  positive  notion,  of  absolute  or  of  infinite  space, 
they  assume  not  only  what  is  not  contained  in 
the  phenomenon,  nay,  they  assume  what  is 
the  very  reverse  of  what  the  phenomenon  man- 
ifests. It  is  plain  that  space  must  either  be 
bounded  or  not  bounded.  These  are  contra- 
dictory alternatives;  on  the  principle  of  Con- 
tradiction they  cannot  both  be  true,  and,  on 
the  principle  of  Excluded  Middle,  one  must 
be  true.  This  cannot  be  denied  without  de- 
nying the  primary  laws  of  intelligence.  But 
though  space  must  be  admitted  to  be  neces- 
sarily either  finite  or  infinite,  we  are  able  to 
conceive  the  possibility  neither  of  its  finitude 
nor  of  its  infinity. 

"  We  are  altogether  unable  to  conceive  space 
as  bounded, — as  finite ;  that  is,  as  a  whole,  be- 
yond which  there  is  no  further  space.  Every 
one  is  conscious  that  this  is  impossible.  It  con- 
tradicts also  the  supposition  of  space  as  a  neces- 


36        THE   CONCEPTION   OF   THE   INFINITE. 

sary  notion ;  for  if  we  could  imagine  space  as 
a  terminated  sphere,  and  that  sphere  not  itself 
enclosed  in  a  surrounding  space,  we  should  not 
be  obliged  to  think  everything  in  space;  and, 
on  the  contrary,  if  we  did  imagine  this  termi- 
nated sphere  as  itself  in  space,  in  that  case  we 
should  not  have  actually  conceived  all  space  as 
a  bounded  whole.  The  one  contradictory  is 
thus  found  inconceivable;  we  cannot  conceive 
space  as  positively  limited. 

"  On  the  other  hand,  we  are  equally  power- 
less to  realize  in  thought  the  possibility  of  the 
opposite  contradictory;  we  cannot  conceive  space 
as  infinite,  as  without  limits.  You  may  launch 
out  in  thought  beyond  the  solar  walk,  you  may 
transcend  in  fancy  even  the  universe  of  matter, 
and  rise  from  sphere  to  sphere  in  the  region  of 
empty  space,  until  imagination  sinks  exhausted; 
with  all  this,  what  have  you  done?  You  have 
never  gone  beyond  the  finite;  you  have  attained, 
at  best,  only  to  the  indefinite,  and  the  indefinite, 
however  expanded,  is  still  always  the  finite.  .  .  . 
Now,  then,  both  contradictories  are  equally  in- 
conceivable, and,  could  we  limit  our  attention  to 
one  alone,  we  should  deem  it  at  once  impossible 
and  absurd,  and  suppose  its  unknown  opposite 


THE  ANTINOMIES   OF  HAMILTON.  37 

as  necessarily  true.  But  as  we  not  only  can,  but 
are,  constrained  to  consider  both,  we  find  that 
both  are  equally  incomprehensible;  and  yet, 
though  unable  to  view  either  as  possible,  we  are 
forced  by  a  higher  law  to  admit  that  one,  but 
one  only,  is  necessary." 

It  is  evident  that  the  difficulties  in  which  Sir 
William  has  here  involved  himself  are  gratu- 
itous. The  argument  used  to  prove  the  latter 
of  the  contradictories  inconceivable  breaks  down 
upon  a  careful  examination. 

We  may  indeed  "  rise  from  sphere  to  sphere 
in  the  region  of  empty  space"  without  trans- 
cending the  finite,  and  the  attempt  to  thus 
transcend  it  is  as  hopeless  as  would  be  the  at- 
tempt of  the  peacock  to  escape  from  his  feet  by 
flying;  for  we  cannot  arrive  at  the  unlimited 
while  we  carry  our  limits  with  us.  Each  suc- 
cessive stage  simply  places  the  limits  farther 
apart,  and  in  no  respect  helps  us  to  do  away 
with  them  altogether. 

Such  a  mode  of  procedure  forcibly  reminds 

one  of  the  amusing  person  in  Chamisso's  poem, 

who  supposed  that  by  quickly  turning  himself 

around  he  could   cause   his   queue  to  hang  in 

front : 

4 


38       THE   CONCEPTION  OF  THE  INFINITE. 

11  Er  dreht  sich  links,  er  dreht  sich  rechts, 
Es  thut  nichts  Gut's,  es  thut  nichts  Schlecht's, 
Der  Zopf,  der  hangt  ihm  hinten." 

And  how  analogous  is  the  condition  of  one 
who  thinks  that  the  way  to  reach  the  infinite  is 
to  endlessly  continue  this  hopeless  journey  be- 
yond "  the  universe  of  matter"  to  that  of  the 
hero  as  portrayed  in  the  last  verse : 

"  Und  seht,  er  dreht  sich  immer  noch 
Und  denkt :  es  hilft  am  Ende  doch — 
Der  Zopf,  der  hangt  ihm  hinten." 

It  is  not  by  adding  space  to  space  that  we 
arrive  at  the  idea  of  infinite  space,  and  imagi- 
nation may  well  sink  exhausted  in  the  attempt 
to  find  the  end  of  the  endless. 

But  to  think  space  as  infinite  it  is  by  no 
means  necessary  to  take  this  journey;  and,  so 
far  from  proving  that  we  cannot  regard  our 
notion  of  space  as  infinite,  the  failure  of  any 
such  attempt  to  know  it  as  a  whole  is  the  surest 
evidence  that  it  is  indeed  infinite.  The  latter 
of  the  contradictories  is  thus  found  to  be  incon- 
ceivable only  when  we  suppose  a  quantitative 
element  in  our  conception  of  infinity,  and,  this 
error  corrected,  the  antinomy  disappears. 


THE  ANTINOMIES   OF  HAMILTON.  39 

Sir  "William  has  applied  the  Law  of  the  Con- 
ditioned also  to  the  minimum  of  space:  "That 
the  conceivable,"  he  continues,  "  lies  always  be- 
tween two  inconceivable  extremes  is  illustrated 
by  every  other  relation  of  thought.  "We  have 
found  the  maximum  of  space  incomprehensible; 
can  we  comprehend  its  minimum?  This  is 
equally  impossible.  Here,  likewise,  we  recoil 
from  one  inconceivable  contradictory  only  to 
infringe  upon  another.  Let  us  take  a  portion 
of  space,  however  small ;  we  can  never  conceive 
it  as  the  smallest.  It  is  necessarily  extended, 
and  may,  consequently,  be  divided  into  a  half  or 
quarters,  and  each  of  these  halves  or  quarters 
may  again  be  divided  into  other  halves  or  quar- 
ters, and  this  ad  infinitum.  But  if  we  are  unable 
to  construe  to  our  mind  the  possibility  of  an 
absolute  minimum  of  space,  we  can  as  little 
represent  to  ourselves  the  possibility  of  an  in- 
finite divisibility  of  any  extended  entity." 

This  is  at  bottom  a  repetition  of  the  above- 
mentioned  error.  Whether  we  regard  space, 
with  the  Kantian,  as  in  its  nature  wholly  com- 
posite, and  always  capable  of  further  subdivision ; 
or,  with  the  Berkeleyan,  as  composed  of  minima 
visibilia,    themselves    not    admitting    of    subdi- 


40       THE   CONCEPTION  OF  THE  INFINITE. 

vision ;  in  neither  case  are  we  forced  into  a 
choice  of  two  inconceivables.  The  Berkeleyan 
would  claim  that  the  difficulty  of  conceiving  a 
component  part  of  any  extended  entity  as  itself 
non-extended  arises  from  the  fact  that  the  min- 
imum visibile  is  represented  by  the  imagination 
as  extended,  the  notion  of  extension  being  car- 
ried over  from  those  objects  to  which  it  right- 
fully belongs,  and  is,  consequently,  not  a  true 
minimum  visibile.  "While  the  Kantian  may  main- 
tain that  there  is  nothing  inconceivable  in'  an 
infinite  series,  rightly  understood.  If  we  sup- 
pose a  series  to  be  infinite,  we  cannot,  of  course, 
represent  it  to  the  mind  as  a  completed  whole, 
but  that  is  unnecessary  to — it  is  incompatible 
with — our  recognition  of  it  as  infinite.  The  an- 
tinomy, like  its  predecessor,  disappears  as  soon 
as  it  is  recognized  that  there  is  no  quantitative 
element  in  our  conception  of  the  infinite.* 

*  It  is  odd  that  the  statement  that  we  are  unable  to  con- 
ceive of  any  portion  of  space  as  the  smallest  possible,  and 
not  itself  divisible  into  spaces,  should  be  so  constantly 
allowed  to  pass  unchallenged.  The  question  is  not  whether, 
when  we  have  carried  our  subdivision  of  a  surface  so  far 
that  an  apparently  unextended  point  of  color  alone  is  left, 
we  can  in  imagination  substitute  for  that  point  an  extended 


THE  ANTINOMIES   OF  HAMILTON.  41 

Let  us  turn  now  to  Sir  William's  application 
of  the  law  to  our  conception  of  time : 

"In  like  manner  Time; — this  is  a  notion 
even  more  universal  than  space,  for  while  we 
exempt  from  occupying  space  the  energies  of 
mind,  we  are  unable  to  conceive  these  as 
not  occupying  time.  Thus,  we  think  every- 
thing, mental  and  material,  as  in  time,  and  out 
of  time  we  can  think  nothing.  But  if  we  at- 
tempt to  comprehend  time,  either  in  whole  or  in 

surface,  and  proceed  to  subdivide  that,  convinced  that  any 
system  of  relations  developed  from  the  latter  may  lawfully 
be  carried  over  to  all  possible  future  experiences  of  a  simi- 
lar nature  which  may  be  found  to  be  connected  with  the 
former ;  but  the  question  really  is  whether  we  can  conceive 
that  identical  apparently  unextended  spot  of  color  divided 
and  subdivided.  If  we  say  that  we  are  dividing  it  in 
thought,  when  our  division  proceeds  thus  through  a  rep- 
resentative, if  we  insist  in  applying  to  the  object  in  the 
two  cases  the  word  "same,"  we  should  never  forget  that 
we  are  using  this  highly  ambiguous  word  "same"  in  by  no 
means  the  strictest  of  its  four  distinctly  different  senses, 
and  should  be  very  sure  that  we  are  not  practising  self- 
deception  by  juggling  with  the  several  meanings  of  the 
word.  When  I  say  that  with  the  Kantian  hypothesis  we 
are  not  forced  to  embrace  what  is  inconceivable,  I  refer 
only  to  the  point  under  consideration,  the  conceivability  of 
an  endless  series  per  se. 

4* 

SIT! 

C4LIF0R^ 


42       THE   CONCEPTION  OF  THE  INFINITE. 

part,  we  find  that  thought  is  hedged  in  between 
two  incomprehensibles.  Let  us  try  the  whole. 
And  here  let  us  look  back, — let  us  consider  time 
a  "parte  ante.  And  here  we  may  surely  flatter 
ourselves  that  we  shall  be  able  to  conceive  time 
as  a  whole,  for  here  we  have  the  past  period 
bounded  by  the  present;  the  past  cannot,  there- 
fore, be  infinite  or  eternal,  for  a  bounded  infi- 
nite is  a  contradiction.  But  we  shall  deceive 
ourselves.  We  are  altogether  unable  to  con- 
ceive time  as  commencing ;  we  can  easily  rep- 
resent to  ourselves  time  under  any  relative 
limitation  of  commencement  and  termination, 
but  we  are  conscious  to  ourselves  of  nothing 
more  clearly,  than  that  it  would  be  equally 
possible  to  think  without  thought,  as  to  con- 
strue to  the  mind  an  absolute  commencement 
or  an  absolute  termination  of  time, — that  is,  a 
beginning  and  an  end  beyond  which  time  is 
conceived  as  non-existent.  Goad  imagination 
to  the  utmost,  it  still  sinks  paralyzed  within 
the  bounds  of  time,  and  time  survives  as  the 
condition  of  the  thought  itself  in  which  we 
annihilate  the  universe.  On  the  other  hand, 
the  concept  of  past  time  as  without  limit — with- 
out commencement — is  equally  impossible.     We 


THE  ANTINOMIES   OF  HAMILTON.  43 

cannot  conceive  the  infinite  regress  of  time; 
for  such  a  notion  could  only  be  realized  by  the 
infinite  addition  in  thought  of  finite  times,  and 
such  an  addition  would  itself  require  an  eternity 
for  its  accomplishment.  If  we  dream  of  effect- 
ing this,  we  only  deceive  ourselves  by  substi- 
tuting the  indefinite  for  the  infinite,  than  which 
no  two  notions  can  be  more  opposed. 

"  The  negation  of  a  commencement  of  time 
involves,  likewise,  the  affirmation  that  an  infinite 
time  has,  at  every  moment,  already  run ;  that  is, 
it  implies  the  contradiction  that  an  infinite  has 
been  completed.  For  the  same  reasons  we  are 
unable  to  conceive  an  infinite  progress  of  time; 
while  the  infinite  regress  and  the  infinite  prog- 
ress, taken  together,  involve  the  triple  contra- 
diction of  an  infinite  concluded,  of  an  infinite 
commencing,  and  of  two  infinities  not  exclusive 
of  each  other."  * 

The  statement  that  past  time  cannot  be  re- 
garded as  infinite  because  limited  by  the  present 

*  "  Metaphysics,"  New  York,  1880,  p.  529.  See,  also, 
Spencer's  "First  Principles,"  New  York,  1875,  p.  81:  "How 
self-destructive  is  the  assumption  of  two  or  more  Infinites 
is  manifest  on  remembering  that  such  Infinites,  by  limiting 
each  other,  would  become  finite." 


44        THE    CONCEPTION   OF   THE   INFINITE. 

is  based  upon  the  erroneous  supposition  that 
what  is  limited  at  one  point  cannot  be  infinite. 
But,  as  has  been  shown  in  the  preceding  chap- 
ter, one  point  is  not  sufficient  to  define  a  line 
as  finite ;  and  time,  which  we  represent  to  our- 
selves under  the  form  of  a  continuous  line,  must 
be  regarded  as  infinite  unless  limited  at  two 
points.  Time  past  and  time  future  are  two  in- 
finites, and  as  such  are  perfectly  conceivable. 
The  difficulty  respecting  the  possibility  of  two 
infinites  mutually  exclusive  of  each  other  is  a 
difficulty  only  under  a  false  conception  of  the 
infinites  as  quantitative  wholes,  and  may  easily 
be  made  to  disappear. 

The  assertion,  also,  that  the  past  cannot  be 
infinite,  as  "  a  bounded  infinite  is  a  contradic- 
tion/' may  well  be  scanned.  Arguments  drawn 
from  the  etymological  signification  of  a  word 
are  of  small  value,  unless  that  expresses  the 
true  and  whole  content  of  the  word.  That  such 
is  not  the  case  here  is  evident.  A  line  limited 
at  but  one  point  is  certainly  not  finite,  for  it 
cannot  be  regarded  as  a  quantitative  whole; 
cannot  be  increased,  diminished,  or  compared 
in  length  with  other  lines;  in  short,  is  not  sub- 
ject to  the  conditions  of  the  finite.     If,  then,  for 


THE  ANTINOMIES   OF  HAMILTON.  45 

etymological  reasons,  we  exclude  it  from  the 
class  of  infinites,  we  have  the  finite,  the  infinite, 
and  a  third  class,  a  terthtm  quid,  which  lies  be- 
tween the  two,  and  might  be  humorously  de- 
scribed as  "  infinite  at  one  end."  But,  etymol- 
ogy aside,  there  is  no  difficulty  in  classing  such 
a  line  with  one  that  has  no  limits,  for  they  are 
subject  to  precisely  the  same  conditions,  and 
equally  distinct  from  the  finite.  However,  the 
appellation  is  a  matter  of  taste ;  the  thing  which 
it  is  important  to  bear  in  mind  is  that  a  line 
with  but  one  limit  can  no  more  be  regarded  as 
a  quantitative  whole  than  a  line  absolutely  with- 
out limits ;  and,  whether  we  choose  to  call  time 
past  infinite  or  finite,  that  we  may  have  a  clear 
knowledge  of  it  as  limited  only  by  the  present, 
without  attempting  to  pass  over  its  parts  in 
succession,  and  thus  arrive  at  the  whole. 

The  assertion,  too,  that  the  negation  of  a 
commencement  of  time  "  implies  the  contradic- 
tion that  an  infinite  has  been  completed"  is 
misleading.  The  word  "  completed"  is  an  un- 
fortunate one  to  use  in  this  connection,  as  it 
suggests  to  the  mind  the  idea  of  progression 
from  a  beginning  to  an  end.  The  denial  of  a 
commencement  of  time  does  imply  that  an  in- 


46        THE   CONCEPTION  OF   THE  INFINITE. 

finite  is  past,  but  not  that  it  is  completed  in  any 
such  sense  as  to  be  enclosed  within  limits,  for 
it  is  quite  conceivable  that  it  never  began,  and 
the  present  moment  is,  by  supposition,  the  only 
limit. 

Removing  the  misconceptions  just  noticed,  the 
whole  force  of  the  argument  for  the  inconceiv- 
ability of  time  as  infinite  lies,  as  in  the  former 
cases,  in  the  idea  that  the  infinite  may  only  be 
known  by  exhausting  it  in  its  totality,  through 
the  successive  addition  of  its  finite  parts,  and 
this  antinomy  also  proves  to  be  a  gratuitous 
one. 

Having  argued  thus  far  the  inconceivability 
of  time  as  a  maximum,  Sir  William  turns,  as  in 
his  discussion  on  our  knowledge  of  space,  to  a 
consideration  of  time  as  a  minimum: 

"  Now  take  the  parts  of  time, — a  moment,  for 
instance;  this  we  must  conceive  as  either  divis- 
ible to  infinity,  or  that  it  is  made  up  of  certain 
absolutely  smallest  parts.  One  or  other  of  these 
contradictories  must  be  the  case.  But  each  is, 
to  us,  equally  inconceivable.  Time  is  a  pro- 
tensive  quantity,  and,  consequently,  any  part  of 
it,  however  small,  cannot,  without  a  contradic- 
tion, be  imagined  as  not  divisible  into  parts,  and 


THE  ANTINOMIES   OF  HAMILTON.  47 

these  parts  into  others  ad  infinitum.  But  the 
opposite  alternative  is  equally  impossible ;  we 
cannot  think  this  infinite  division.  One  is  neces- 
sarily true ;  but  neither  can  be  conceived  pos- 
sible. It  is  on  the  inability  of  the  mind  to  con- 
ceive either  the  ultimate  indivisibility  or  the 
endless  divisibility  of  space  and  time  that  the 
arguments  of  the  Eleatic  Zeno  against  the  pos- 
sibility of  motion  are  founded,  —  arguments 
which  at  least  show  that  motion,  however  cer- 
tain as  a  fact,  cannot  be  conceived  possible,  as 
it  involves  a  contradiction."  * 

With  reference  to  the  former  of  the  alterna- 
tives offered,  the  Berkeleyan  would  answer  that 
in  the  case  of  "  protensive"  quantity,  as  in  the 
case  of  extensive,  the  difficulty  of  conceiving 
the  unit,  itself  indivisible,  lies  in  the  imagina- 
tion, and  may,  with  precautions,  be  obviated; 
while  with  reference  to  the  latter,  the  Kantian 
may  answer,  as  before,  that  an  infinite  series  is 
not  intrinsically  unthinkable.  I  may  remark, 
en  passant,  that  the  idea  of  the  theoretic  impos- 
sibility of  motion  is  a  wholly  erroneous  one,  and 
falls  with  the  errors  upon  which  it  is  based. 

*  "  Metaphysics,"  New  York,  1880,  pp.  529-30. 


48        THE   CONCEPTION   OF  THE  INFINITE. 

One  turns  from  an  examination  of  Sir  Wil- 
liam Hamilton's  application  of  the  Law  of  the 
Conditioned  to  Space  and  Time  with  a  convic- 
tion that  if  the  Philosophy  of  the  Conditioned 
has  no  better  props  to  sustain  it  than  these 
prove  to  be,  it  may  turn  out  as  insecure  an 
edifice  as  the  house  that  a  certain  foolish  indi- 
vidual, according  to  the  parable,  founded  upon 
sand. 

It  remains  to  consider  a  case  which  ap- 
parently militates  against  the  theory  that  an 
infinite  series  can  never  be  regarded  as  a 
whole. 

Let  us  suppose  a  point  moving  uniformly 
along  the  line  AB,  over  the  whole  of  which  it 
can  pass  in  one  minute.  In  J  of  a  minute  it 
will  have  passed  over  J  of  the  line;  in  J  of  a 
minute  over  J  more ;  in  J  of  a  minute  J  more, 
etc.  When  the  minute  is  completed  the  point 
will  have  passed  over  the  whole  line.  Has  it 
not  passed  successively  over  the  whole  series, 
thus  completing  it,  and  arriving  at  0  as  its 
lower  limit?  And  may  we  not  say  that  the 
sum  of  all  the  terms  in  the  series  is  equal  to 
the  whole  line  passed  over  ?  Is  it  not  a  quan- 
titative whole? 


THE  ANTINOMIES  OF  HAMILTON.  49 

A  little  reflection  will  reveal  the  fallacy  in 
this  reasoning.  The  series  is  not  completed  at 
all,  but  is  truly  infinite.  It  is  limited  at  one 
poiut  by  the  highest  term  (J),  but  is  not  limited 
at  another  point  by  a  lowest  term  (0) ;  for  the  0 
can  only  be  assumed  as  a  limit  to  the  series  by 
breaking  the  law  of  the  series,  which  is  that 
each  term  shall  be  half  as  great  as  the  one  pre- 
ceding. 

We  can  never,  by  halving  something,  arrive 
at  nothing;  a  division  of  substance  will  never 
give  us  that  which  is  not  substance.  The  0, 
since  it  does  not  make  one  in  the  series,  cannot 
limit  the  series.  The  Kantian  may  answer  the 
question  by  reasoning  as  follows :  "  The  point 
in  question  passes  over  the  whole  line,  not  by 
completing  the  descending  series  until  it  arrives 
at  a  lowest  term  in  the  simple,  and  from  that 
passes  to  zero,  but  by  the  successive  addition  of 
spaces,  which  are  themselves  composites.  '  As 
space  is  not  a  composite  of  substances  (and  not 
even  of  real  accidents),  if  I  abstract  all  compo- 
sition therein,  nothing,  not  even  a  point,  re- 
mains ;  for  a  point  is  possible  only  as  the  limit 
of  a  space,  consequently  of  a  composite.  Space 
and  time,  therefore,  do   not   consist  of  simple 

5 


50       THE  CONCEPTION  OF  THE  INFINITE. 

parts.'  *  We  cannot,  therefore,  consider  any 
member  of  the  series  in  question  as  the  smallest 
possible,  nor  the  zero  as  a  limit  to  the  series; 
nor  can  we  regard  the  series  as  in  any  sense 
completed.  If  we  discontinue  the  subdivision 
at  any  point  whatever,  we  may  justly  say  that 
the  foot  or  yard  contains  all  the  terms  of  the 
series.  But  when  the  point  has  reached  zero  it 
has  reached  it  by  breaking  the  series,  not  com- 
pleting it.  A  completion  the  law  of  the  series 
renders  impossible."  f  And  the  Berkeleyan 
might  answer  that  a  completion  of  the  journey 
along  the  line  by  no  means  implies  the  com- 

*  "  Critique  of  Pure  Reason  ;"  "Observations  on  the  Sec- 
ond Antinomy."     Ed.  Bohn. 

f  How  the  series  can  be  broken  by  the  progressive  motion 
of  a  point  over  a  line  when,  by  hypothesis,  the  series  is 
throughout  applicable  to  the  line  (i.e.,  the  line  is  infinitely 
divisible),  the  Kantian  must  explain ;  the  fact  remains  that 
neither  an  infinite  nor  a  finite  series  can  be  completed  by 
breaking  its  law;  and  unless  we  claim  that  half  of  some- 
thing, when  that  something  is  small  enough,  is  equal  to 
nothing,  we  cannot  bring  the  zero  into  the  series  as  a  limit. 
Even  could  this  be  done,  the  desired  point  would  not  be 
established,  as  the  series  would  then  be  limited  at  two  points, 
consequently  in  no  sense  infinite,  an,d  its  completion  would 
not  be  the  completion  of  an  infinite  series. 


THE  ANTINOMIES   OF  HAMILTON.  51 

pletion  of  an  infinite  series,  since  relations  of 
quantity  may  be  divorced  from  all  content  of 
actual  being,  and  used  purely  symbolically. 
Mathematical  reasonings,  he  would  argue,  are 
applicable  to  space  only  within  the  limits  of  a 
possible  perception  ;  and  the  series  may  be  truly 
infinite,  though  the  line-portions  to  which  some 
of  its  members  are  applied  may  be  limited  in 
number.* 

This  problem,  it  will  be  seen,  is  simply  the 
old  puzzle  of  Achilles  and  the  tortoise  in  a 
somewhat  altered  dress ;  and  the  answer  which 
has  been  given  to  that  puzzle — that  the  series 
"  runs  into  infinitesimals  which  are  practical 
zeros,  and,  even  if  theoretically  infinite  in  num- 
ber, really  are  all  included  in  that  finite  length 
which  Achilles  will  quickly  get  over"f — is  not 

*  I  have  taken  up  both  positions  to  bring  out  clearly  the 
fact  that  the  assertion  of  the  conceivability  of  the  infinite 
is  quite  independent  of  metaphysical  theories  as  to  the  nature 
of  space  and  time.  The  two  questions  are  wholly  distinct,  as 
I  will  show  in  a  later  chapter. 

f  This  is  the  solution  of  the  problem  given  by  the  late  Pro- 
fessor Atwater  in  his  little  book  on  the  Elements  of  Logic. 
The  explanation  offered  could  not  be  much  worse.  What, 
we  may  ask,  is  meant  by  a  "  practical  zero,"  as  distinguished 
from  a  theoretical?     And  the  assertion  that  these  practical 


52       THE   CONCEPTION   OF  THE  INFINITE. 

a  true  answer,  since  it  regards  a  series,  which, 
by  the  very  terms  of  its  statement,  is  incapable 
of  completion,  as  a  completed  whole  and  equal 
to  a  finite. 

Before  leaving  this  chapter  I  would  remark 
that  it  is  interesting  to  notice  how  wide-spread 
has  been  the  conviction  that  the  only  way  to 
arrive  at  a  cognition  of  the  infinite  is  to  proceed 
to  an  endless  addition  of  finites.  Hamilton,  we 
have  seen,  makes  much  of  it.  Mansel  follows 
in  his  footsteps.  Mr.  Herbert  Spencer  quotes 
him  with  approbation.  And  Kant,  as  we  shall 
see  in  the  chapter  following,  did  not  escape  the 
snare.  "  To  have  actually  in  the  mind  the  idea 
of  a  space  infinite,"  says  Locke,*  "  is  to  suppose 
the  mind  already  passed  over,  and  actually  to 
have  a  view  of  all  those  repeated  ideas  of  space 
which  an  endless  repetition  can  never  totally 
represent  to  it; — which  carries  in  it  a  plain  con- 
tradiction." 

zeros,  "even  if  theoretically  infinite  in  number,  really  are 
all  included"  in  a  finite  length,  would  seem  to  draw  a  dis- 
tinction between  the  theoretical  and  the  practical,  little  to 
the  advantage  of  the  former.  It  makes  theoretical  about 
equivalent  to  unreal. 

*  "Essay  concerning  Human  Understanding,"  book  ii. 
ch.  xvii.,  I  7. 


CHAPTER    IV. 

KANT,  MILL,  AND   CLIFFORD. 

It  is  interesting  to  notice  that  the  truth  that 
our  conception  of  infinity  contains  no  quanti- 
tative element  has  been  seen,  like  Thule, 
"  through  the  mist"  by  several  acute  minds, 
who  have  yet  not  seen  the  truth  with  sufficient 
clearness  to  escape  the  common  errors  arising 
from  the  introduction  of  the  contradictory  ele- 
ment into  their  discussions.  Immanuel  Kant, 
although  he  has  based  the  proof  of  the  thesis  of 
his  first  antinomy  on  a  false  conception  of  in- 
finity, and  although,  after  correctly  criticising 
the  false  conception,  he  himself  lapses  into  it, 
yet  perceived,  and  in  so  many  words  gave  ex- 
pression to  the  fact,  that  the  conception  of  the 
infinite  is  qualitative. 

The  thesis  of  the  first  antinomy  maintains 
that  the  world  has  a  beginning  in  time,  and  is 
limited  with  regard  to  space,  both  of  which 
propositions  are  denied  in  the  antithesis.  The 
proofs  offered  in  support  of  the  antithesis  may 

5*  53 


54       THE  CONCEPTION  OF  THE  INFINITE. 

be  passed  over  as  extraneous  to  the  subject ; 
those  in  support  of  the  thesis  I  will  quote,  not 
for  the  purpose  of  again  refuting  them,  for  they 
are  identical  with  those  used  by  Sir  William 
Hamilton  in  his  antinomies,  but  that  I  may 
give  the  observations  appended  to  them,  which 
are  very  significant  in  their  contextual  connec- 
tion. The  proof  proceeds  by  assuming  the 
truth  of  the  antithesis,  and  then  showing  it  to 
be  impossible. 

"  Granted,  that  the  world  has  no  begin- 
ning in  time;  up  to  every  given  moment  of 
time  an  eternity  must  have  elapsed,  and  there- 
with passed  away  an  infinite  series  of  suc- 
cessive conditions  or  states  of  things  in  the 
world.  Now,  the  infinity  of  a  series  consists 
in  the  fact  that  it  never  can  be  completed 
by  means  of  a  successive  synthesis.  It  fol- 
lows that  an  infinite  series,  already  elapsed, 
is  impossible,  and  that  consequently  a  begin- 
ning of  the  world  is  a  necessary  condition  of 
its  existence.  And  this  was  the  first  thing  to 
be  proved. 

"  As  regards  the  second,  let  us  take  the  oppo- 
site for  granted.  In  this  case  the  world  must 
be  an  infinite  given  total  of  coexistent  things. 


KANT,  MILL,  AND  CLIFFORD.  55 

Now,  we  cannot  cogitate  the  dimensions  of  a 
quantity,  which  is  not  given  within  certain  lim- 
its of  an  intuition,  in  any  other  way  than  by 
means  of  the  synthesis  of  its  parts,*  and  the 
total  of  such  a  quantity  only  by  means  of  a 
completed  synthesis  or  the  repeated  addition  of 
unity  to  itself.  Accordingly,  to  cogitate  the 
world,  which  fills  all  spaces,  as  a  whole,  the 
successive  synthesis  of  the  parts  of  an  infinite 
world  must  be  looked  upon  as  completed, — that 
is  to  say,  an  infinite  time  must  be  regarded  as 
having  elapsed  in  the  enumeration  of  all  co- 
existing things,  which  is  impossible.  For  this 
reason  an  infinite  aggregate  of  actual  things 
cannot  be  considered  as  a  given  whole,  conse- 
quently, not  as  a  contemporaneously  given 
whole.  The  world  is  consequently,  as  regards 
extension  in   space,  not  infinite,  but  enclosed  in 

*  Kant  says,  in  a  foot-note,  "  We  may  consider  an  unde- 
termined quantity  as  a  whole  when  it  is  enclosed  within 
limits,  although  we  cannot  construct  or  ascertain  its  totality 
by  measurement, — that  is,  by  the  successive  synthesis  of  its 
parts.  For  its  limits  of  themselves  determine  its  complete- 
ness as  a  whole.'1'  This  method  being  absent  in  the  case  of 
infinites,  Kant  thinks  he  can  cognize  them  only  by  falling 
back  upon  the  successive  synthesis. 


56       THE   CONCEPTION  OF  THE  INFINITE. 

limits.     And  this  was  the  second   thing  to  be 
proved."* 

It  will  be  noticed  that  in  the  first  part  of  the 
proof  the  word  completed  (vollendet)  is  used  in 
the  manner  before  objected  to  as  misleading. 
When  we  speak  of  a 'series  as  "completed  by 
means  of  a  successive  synthesis,"  we  are  apt 
to  regard  it  as  a  whole,  with  a  beginning  as 
well  as  an  end.  Of  course,  when  we  are  con- 
sidering time  past  as  limited  by  the  present, 
such  an  association  must  be  unfortunate. 

The  observations  on  the  thesis  are  the  fol- 
lowing : 

"  In  bringing  forward  these  conflicting  argu- 
ments, I  have  not  been  on  the  search  for  soph- 
isms for  the  purpose  of  availing  myself  of 
special  pleading,  which  takes  advantage  of  the 
carelessness  of  the  opposite  party,  appeals  to  a 
misunderstood  statute,  and  erects  its  unright- 
eous claims  upon  an  unfair  interpretation.  Both 
proofs  originate  fairly  from  the  nature  of  the 
case,  and  the  advantage  presented  by  the  mis- 

*  Iramanuel  Kant's  "  Sammtliche  Werke,"  Leipzig,  1867. 
Dritter  Band,  s.  304.  I  have  taken  the  rendering  of  Meikle- 
john's  excellent  translation  of  the  Critique. 


KANT,  MILL,  AND  CLIFFORD.  57 

takes  of  the  dogmatists  of  both  parties  has  been 
completely  set  aside. 

"  The  thesis  might  also  have  been  unfairly 
demonstrated  by  the  introduction  of  an  erro- 
neous conception  of  the  infinity  of  a  given 
quantity.  A  quantity  is  infinite  if  a  greater 
than  itself  cannot  possibly  exist.  The  quantity 
is  measured  by  the  number  of  given  units — 
which  are  taken  as  a  standard — contained  in 
it.  Now,  no  number  can  be  the  greatest,  be- 
cause one  or  more  units  can  always  be  added. 
It  follows  that  an  infinite  given  quantity,  conse- 
quently an  infinite  world  (both  as  regards  time 
and  extension),  is  impossible.  It  is,  therefore, 
limited  in  both  respects.  In  this  manner  I 
might  have  conducted  my  proof;  but  the  con- 
ception given  in  it  does  not  agree  with  the  true 
conception  of  an  infinite  whole.  In  this  there 
is  no  representation  of  its  quantity ;  it  is  not 
said  how  large  it  is ;  consequently  its  concep- 
tion is  not  the  conception  of  a  maximum.  "We 
cogitate  in  it  merely  its  relation  to  an  arbitra- 
rily assumed  unit,  in  relation  to  which  it  is 
greater  than  any  number.  Now,  just  as  the 
unit  which  is  taken  is  greater  or  smaller,  the 
infinite  will  be  greater  or  smaller;  but  the  in- 


58        THE   CONCEPTION  OF   THE  INFINITE. 

finity,  which  consists  merely  in  the  relation  to 
this  given  unit,  must  remain  always  the  same, 
although  the  absolute  quantity  of  the  whole  is 
not  thereby  cognized. 

"  The  true  (transcendental)  conception  of  in- 
finity is  that  the  successive  synthesis  of  unity  in 
the  measurement  of  a  given  quantum  can  never 
be  completed.  Hence  it  follows,  without  possi- 
bility of  mistake,  that  an  eternity  of  actual  suc- 
cessive states  up  to  a  given  (the  present)  mo- 
ment cannot  have  elapsed,  and  that  the  world 
must  therefore  have  a  beginning. 

"In  regard  to  the  second  part  of  the  thesis, 
the  difficulty  as  to  an  infinite  and  yet  elapsed 
series  disappears ;  for  the  manifold  of  a  world 
infinite  in  extension  is  contemporaneously  given. 
But,  in  order  to  cogitate  the  total  of  this  mani- 
fold, as  we  cannot  have  the  aid  of  limits  consti- 
tuting by  themselves  this  total  in  intuition,  we 
are  obliged  to  give  some  account  of  our  concep- 
tion, which  in  this  case  cannot  proceed  from 
the  whole  to  the  determined  quantity  of  the 
parts,  but  must  demonstrate  the  possibility  of  a 
whole  by  means  of  a  successive  synthesis  of  the 
parts.  But  as  this  synthesis  must  constitute  a 
series  that  cannot  be  completed,  it  is  impossible 


KANT,  MILL,  AND  CLIFFORD.  59 

for  us  to  cogitate  prior  to  it,  and  consequently 
not  by  means  of  it,  a  totality.  For  the  concep- 
tion of  totality  itself  is  in  the  present  case  the 
representation  of  a  completed  synthesis  of  the 
parts;  and  this  completion,  and  consequently  its 
conception,  is  impossible." 

We  here  find  brought  forward  a  conception 
of  the  infinite  which  is  declared  faulty;  a  dec- 
laration of  the  point  in  which  it  differs  from  the 
true  conception;  and  a  statement  of  what,  ac- 
cording to  Kant,  the  true  conception  really  is. 
The  false  conception  is  that  "a  quantity  is  in- 
finite if  a  greater  than  itself  cannot  possibly 
exist."  We  may  readily  see  that  such  a  con- 
ception gives  us,  not  an  infinite,  but  a  finite. 
Not  only  is  the  word  "  greater"  inapplicable  to 
infinites,  but  the  very  expression,  "  a  quantity  is 
infinite,"  is  absurd,  as  involving  a  flat  contra- 
diction. Kant  was  too  keen  a  thinker  not  to 
see  that  that  which  admits  of  an  addition  ot 
units,  and,  consequently,  of  increase  as  a  whole, 
cannot  be  infinite.  This  does  not  agree,  he  says, 
with  the  true  conception  of  the  infinite,  in  which 
"there  is  no  representation  of  its  quantity;  it  is 
not  said  how  large  it  is;  consequently  its  con- 
ception is  not  the  conception  of  a  maximum." 


GO        THE   CONCEPTION  OF   THE  INFINITE. 

Could  there  be  a  clearer  recognition  of  the  fact 
that  the  conception  is  not  quantitative  ? 

But  it  is  evident  that  Kant  did  not  see  the 
full  force  and  the  logical  consequences  of  the 
statement.  In  the  sentence  immediately  pre- 
ceding the  one  in  which  he  recognizes  the  quali- 
tative character  of  the  conception  he  uses  the 
phrase  "an  infinite  whole;"  and  in  the  sentences 
immediately  following  he  brings  forward  a  con- 
ception faulty  in  precisely  the  same  respect  as  the 
one  criticised :  "  We  cogitate  in  it  merely  its  rela- 
tion to  an  arbitrarily  assumed  unit,  in  relation  to 
which  it  is  greater  than  any  number.  Now,  just 
as  the  unit  which  is  taken  is  greater  or  smaller, 
the  infinite  will  be  greater  or  smaller ;  but  the 
infinity,  which  consists  merely  in  the  relation  to 
this  given  unit,  must  remain  always  the  same, 
although  the  absolute  quantity  of  the  whole  is 
not  thereby  cognized."  That  is,  if  we  designate 
the  infinite  by  a,  the  unit  by  6,  and  the  infinity 
(the  relation  between  a  and  b)  by  x,  we  find  that 
a  varies  as  b,  and  x  remains  always  the  same  (this 
can  only  mean  numerically  the  same).  In  this 
case  x  is  simply  an  indefinite  number,  and  the 
"  absolute  quantity  of  the  whole"  can  certainly 
be  cognized.     When  we  say  the  infinity  remains 


KANT,  MILL,  AND   CLIFFORD.  Q\ 

always  the  same,  the  question  naturally  arises, 
The  same  in  what?  In  amount?  If  so,  we 
have  but  the  finite.  The  error  is  here,  perhaps, 
not  quite  so  palpable,  but  is  just  as  real  as  in 
the  case  which  Kant  criticises,  and  it  is  of  pre- 
cisely the  same  nature.  Both  parts  of  the  proof 
given  in  support  of  the  thesis  of  course  fall  to 
the  ground  when  this  error  is  rectified. 

The  last  two  observations  are  merely  a  re- 
statement of  the  proofs  of  the  thesis.  The  re- 
mark made  in  the  last  one,  that  "in  order  to 
cogitate  the  total  of  this  manifold,  as  we  cannot 
have  the  aid  of  limits  constituting  by  themselves 
this  total  in  intuition,  we  are  obliged  to  give 
some  account  of  our  conception,  which  in  this 
case  cannot  proceed  from  the  whole  to  the  de- 
termined quantity  of  the  parts,  but  must  dem- 
onstrate the  possibility  of  a  whole  by  means  of 
a  successive  synthesis  of  the  parts,"  will  lose 
all  its  force  if  the  words  "  total"  and  "  whole" 
are  abstracted.  If  this  manifold  is  to  be  known 
as  a  whole,  we  cannot,  of  course,  arrive  at  a 
knowledge  of  it  without  cogitating  all  its  parts 
as  contained  in  it;  but  since  it  is  impossible 
that  it  should  be  thus  known,  we  may  "give 
some    account    of    our    conception"   by   simply 


(52       THE   CONCEPTION  OF  THE  INFINITE. 

stating  that  it  is  without  limit,  thus  recognizing 
it  as  truly  infinite. 

It  seems  odd  that  Kant  should  have  seen  an 
error,  objected  to  it,  and  fallen  into  it  on  the 
same  page;  should  have  said  that  there  is  no 
representation  of  its  quantity  in  the  conception 
of  the  infinite,  and  then  have  called  it  a  whole; 
declared  that  the  true  conception  does  not  say 
how  large  it  is,  and  yet  have  affirmed  that  the 
infinity  is  always  the  same,  while  the  series  as  a 
whole  varies  with  the  unit.  But,  with  and  in 
spite  of  all  his  inconsistencies,  it  must  be  allowed 
that  he  recognized  the  truth,  however  imper- 
fectly, that  our  conception  of  the  infinite  has, 
properly  speaking,  no  quantitative  element,  but 
is  purely  qualitative. 

This  truth  has  also  been  recognized,  though 
less  clearly,  by  John  Stuart  Mill,  in  his  "Ex- 
amination of  Sir  William  Hamilton's  Philoso- 
phy," where  he  maintains,  in  opposition  to  Sir 
William,  that  the  infinite  is  not  inconceivable ; 
and  we  can  see  in  the  paragraphs  which  he 
devotes  to  the  subject  that  Mill  had  in  mind 
two 'distinct  conceptions,  a  quantitative  and  a 
qualitative.  To  the  former,  which  he  calls  the 
adequate  conception,  he  acknowledges  we  cannot 


KANT,  MILL,  AND  CLIFFORD.  (53 

attain ;  the  latter,  which  is  truly  qualitative, 
though  he  does  not  apply  to  it  that  name,  he 
claims  to  be  a  real  conception,  implying  nothing 
inconceivable. 

His  attempted  refutation  of  Hamilton's  argu- 
ment for  the  inconceivability  of  infinite  space 
proceeds  as  follows:* 

"  Our  author  goes  on  to  repeat  the  argument 
used  in  his  reply  to  Cousin,  that  Infinite  Space 
is  inconceivable,  because  all  the  conception  we 
are  able  to  form  of  it  is  negative,  and  a  negative 
conception  is  the  same  as  no  conception.  '  The 
infinite  is  conceived  only  by  the  thinking  away 
of  every  character  by  which  the  finite  was  con- 
ceived.' To  this  assertion  I  oppose  my  former 
reply.  Instead  of  thinking  away  every  char- 
acter of  the  finite,  we  think  away  only  the  idea 
of  an  end,  or  a  boundary.  Sir  W.  Hamilton's 
proposition  is  true  of  '  The  Infinite,'  the  mean- 
ingless abstraction,  but  it  is  not  true  of  Infinite 
Space.  In  irying  to  form  a  conception  of  that, 
we  do  not  think  away  its  positive  characters. 
"We  leave  to  it  the  character  of  space ;  all  that 

*  "Examination  of  Sir  William  Hamilton's  Philosophy," 
Boston,  1868,  vol.  i.,  pp.  104  et  seq. 


64       THE   CONCEPTION  OF  THE  INFINITE. 

belongs  to  it  as  space;  its  three  dimensions,  with 
all  their  geometrical  properties.  We  leave  to 
it  also  a  character  which  belongs  to  it  as  Infi- 
nite, that  of  being  greater  than  any  other  space. 
If  an  object  which  has  these  well-marked  posi- 
tive attributes  is  unthinkable,  because  it  has  a 
negative  attribute  as  well,  the  number  of  think- 
able objects  must  be  remarkably  small.  Nearly 
all  our  positive  conceptions  which  are  at  all 
complex  include  negative  attributes.  I  do  not 
mean  merely  the  negatives  which  are  implied  in 
affirmatives,  as  in  saying  that  snow  is  white  we 
imply  that  it  is  not  black,  but  independent  neg- 
ative attributes  superadded  to  these,  and  which 
are  so  real  that  they  are  often  the  essential  char- 
acters, or  differentiae,  of  classes.  Our  concep- 
tion of  dumb  is  of  something  which  cannot 
speak ;  of  the  brutes,  as  of  creatures  which  have 
not  reason ;  of  the  mineral  kingdom,  as  the  part 
of  nature  which  has  not  organization  and  life; 
of  immortal,  as  that  which  never  dies.  Are  all 
these  examples  of  the  Inconceivable  ?  So  false 
is  it  that  to  think  a  thing  under  a  negation  is  to 
think  it  as  unthinkable. 

"  In  other  passages,  Sir  "W.  Hamilton  argues 
that  we  cannot  conceive  infinite  space,  because 


KANT,  MILL,  AND  CLIFFORD.  65 

we  would  require  infinite  time  to  do  it  in.  It 
would  of  course  require  infinite  time  to  carry 
our  thoughts  in  succession  over  every  part  of  in- 
finite space.  But  on  how  many  of  our  finite 
conceptions  do  we  think  it  necessary  to  perform 
such  an  operation  ?  Let  us  try  the  doctrine 
upon  a  complex  whole,  short  of  infinite,  such  as 
the  number  695,788.  Sir  W.  Hamilton  would 
not,  I  suppose,  have  maintained  that  this  num- 
ber is  inconceivable.  How  long  did  he  think  it 
would  take  to  go  over  every  separate  unit  of 
this  whole,  so  as  to  obtain  a  perfect  knowledge 
of  that  exact  sum,  as  different  from  all  other 
sums,  either  greater  or  less  ?  Would  he  have 
said  that  we  could  have  no  conception  of  the 
sum  until  this  process  had  been  gone  through  ? 
"We  could  not  indeed  have  an  adequate  concep- 
tion. Accordingly,  we  never  have  an  adequate 
conception  of  any  real  thing.  But  we  have  a 
real  conception  of  an  object  if  we  conceive  it  by 
any  of  its  attributes  that  are  sufficient  to  distin- 
guish it  from  all  other  things.  We  have  a  con- 
ception of  any  large  number  when  we  have 
conceived  it  by  some  one  of  its  modes  of  com- 
position, such  as  that  indicated  by  the  position 
of  its  digits.     We  seldom  get  nearer  than  this 

6* 


QQ        THE   CONCEPTION  OF  THE  INFINITE. 

to  an  adequate  conception  of  any  large  number. 
But  for  all  intellectual  purposes  this  limited 
conception  is  sufficient ;  for  it  not  only  enables 
us  to  avoid  confounding  the  number  in  our  cal- 
culations with  any  other  numerical  whole, — 
even  with  those  so  nearly  equal  to  it  that  no 
difference  between  them  would  be  perceptible 
by  sight  or  touch,  unless  the  units  were  drawn 
up  in  a  manner  expressly  adapted  for  displaying 
it, — but  we  can  also,  by  means  of  this  attribute 
of  the  number,  ascertain  and  add  to  our  concep- 
tion as  many  more  of  its  properties  as  we  please. 
If,  then,  we  can  obtain  a  real  conception  of  a 
finite  whole  without  going  through  all  its  com- 
ponent parts,  why  deny  us  a  real  conception  of 
an  infinite  whole  because  to  go  through  them  all 
is  impossible  ?  Not  to  mention  that  even  in  the 
case  of  the  finite  number,  though  the  units  com- 
posing it  are  limited,  yet,  Number  being  infinite, 
the  possible  modes  of  deriving  any  given  num- 
ber from  other  numbers  are  numerically  infinite; 
and  as  all  these  are  necessary  parts  of  an  ade- 
quate conception  of  any  number,  to  render  our 
conception  even  of  this  finite  whole  perfectly 
adequate  would  also  require  an  infinite  time. 
"  But  though  our  conception  of  infinite  space 


KANT,  MILL,  AND  CLIFFORD.  67 

can  never  be  adequate,  since  we  can  never  ex- 
haust its  parts,  the  conception,  as  far  as  it 
goes,  is  a  real  conception.  We  completely 
realize  in  imagination  the  various  attributes 
composing  it.  "We  realize  it  as  Space.  We 
realize  it  as  greater  than  any  given  space.  Wre 
even  realize  it  as  endless,  in  an  intelligible 
manner, — that  is,  we  clearly  represent  to  our- 
selves that  however  much  of  space  has  been 
already  explored,  and  however  much  more  of 
it  we  may  imagine  ourselves  to  traverse,  we  are 
no  nearer  to  the  end  of  it  than  we  were  at 
first  time ;  however  often  we  repeat  the  process 
of  imagining  distance  extending  in  any  direc- 
tion from  us,  that  process  is  always  susceptible 
of  being  carried  farther.  This  conception  is 
both  real  and  perfectly  definite.  It  is  not  vague 
and  indeterminate  as  a  merely  negative  notion 
is.  We  possess  it  as  completely  as  we  possess 
any  of  our  clearest  conceptions,  and  we  can 
avail  ourselves  of  it  as  well  for  ulterior  mental 
operations.  As  regards  the  Extent  of  Space, 
therefore,  Sir  W.  Hamilton  does  not  seem  to 
have  made  out  his  point;  one  of  the  two  con- 
tradictory hypotheses  is  not  inconceivable." 
One   can   see    from   these    extracts    how  the 


68        THE   CONCEPTION  OF  THE  INFINITE. 

idea  of  quantity  entered  into  and  vitiated 
Mill's  reasonings  on  the  infinite.  In  arguing 
that  our  notion  of  infinite  space  is  not  a  purely 
negative  one,  he  enumerates  several  positive  at- 
tributes that  we  leave  to  the  conception  when  we 
take  away  the  notion  of  limits.  Among  them 
he  places  "a  character  which  belongs  to  it  as 
Infinite,  that  of  being  greater  than  any  other 
space."  Evidently  the  character  of  being  greater 
than  any  other  space  is  a  quantitative  attribute, 
and  can  belong  only  to  "a  space," — a  finite; 
while  space  infinite  is  not,  properly  speaking,  "  a 
space"  at  all.  Since  it  has  no  size,  it  cannot  be 
marked  out  from  other  spaces  by  its  size.  And 
Mill  has  manifestly  committed  the  same  error 
which  has  misled  Sir  William  Hamilton,  whom 
he  criticises,  when  he  says  that  it  would,  .of 
course,  require  infinite  time  to  carry  our  thoughts 
in  succession  over  every  part  of  infinite  space. 
How  can  we  speak  of  every  part  of  that  which 
is  not  quantitatively  considered?  of  that  which, 
by  its  very  definition,  is  incapable  of  being  a 
whole  ?  This  is  precisely  Hamilton's  error,  and 
the  cause  of  all  his  difficulties.  A  little  farther 
on  Mill  speaks, just  as  Kant  does,  of  "an  infi- 
nite whole,"  never  noticing  the  contradiction  in 


KANT,  MILL,  AND  CLIFFORD.  69 

the  adjective;  and  in  the  paragraph  following  he 
repeats  a  former  blunder  in  the  statement  that 
our  knowledge  of  space  can  never  be  adequate, 
"since  we  can  never  exhaust  its  parts." 

It  is  strange  that  side  by  side  with  this  con- 
ception of  a  quantitative  infinite,  which,  to  be 
adequately  known,  must  be  known  as  a  whole, 
we  should  find  a  real,  though  imperfect,  anal- 
ysis of  the  true  conception,  and  an  affirmation 
of  its  conceivability.  We  can  conceive  infinite 
space,  says  Mill ;  we  can  conceive  it  as  space, 
as  greater  than  any  given  space,  and  even  as 
endless,  in  an  intelligible  manner.  When  he 
comes  to  describe  this  intelligible  manner  of 
knowing  the  infinity  of  space,  he  uses  an  un- 
fortunate phrase, — "  we  clearly  represent  to  our- 
selves that  however  much  of  space  has  been 
already  explored,  ...  we  are  no  nearer  to 
the  end  of  it  than  we  were  at  first  time;" 
dragging  in  the  idea  of  a  limit,  which  is  also 
done,  as  above  remarked,  by  calling  it  greater 
than  any  other  space.  But  with  all  these  con- 
cessions to  the  old  erroneous  doctrine,  it  cannot 
be  denied  that  Mill  held  that  we  may  know  In- 
finite Space  in  some  other  way  than  by  a  suc- 
cessive  synthesis    of  finite  spaces,  and  that  he 


70       THE   CONCEPTION  OF  THE  INFINITE. 

attempted  an  enumeration  of  the  psychical  ele- 
ments comprehended  by  the  conception,  leaving 
out  the  notion  of  quantity.  This  conception, 
which  he  calls  inadequate,  but  which  he  yet 
insists  upon  as  a  conception  of  the  infinite,  is 
qualitative,  and  harmonizes  with  the  true  char- 
acter of  our  conception  of  infinity. 

The  last  writer  to  whom  I  will  advert  as 
having  had  a  knowledge,  more  or  less  clear,  of 
the  true  qualitative  nature  of  our  conception 
of  infinity,  is  Professor  William  Kingdon  Clif- 
ford, by  whose  untimely  death  England  has  lost 
one  of  her  acutest  and  most  analytic  minds. 
There  are  in  his  short  paper  entitled  "  Of  Boun- 
daries in  General,"  and  devoted  to  the  point,  the 
line,  and  the  surface,  some  interesting  and  sig- 
nificant passages,  which  I  will  quote,  and  upon 
which  I  will  afterwards  comment.  The  pas- 
sages are  these : 

"Infinite  ;  it  is  a  dreadful  word,  I  know,  until 
you  find  out  that  you  are  familiar  with  the 
thing  which  it  expresses.  In  this  place  it 
means  that  between  any  two  positions  there 
is  some  intermediate  position;  between  that 
and  either  of  the  others,  again,  there  is  some 
other  intermediate ;    and  so  on  without  any  end. 


KANT,  MILL,  AND  CLIFFORD.  71 

Infinite  means  without  any  end.  If  you  went 
on  with  that  work  of  counting  forever,  you 
would  never  get  any  farther  than  the  begin- 
ning of  it.  At  last  you  would  have  two  po- 
sitions very  close  together,  but  not  the  same; 
and  the  whole  process  might  be  gone  over 
again,  beginning  with  those  as  many  times  as 
you  like."  .  .  . 

"  In  fact,  when  we  said  that  there  is  an  infi- 
nite number  of  points  in  a  piece  of  line-room, 
we  might  have  said  a  great  deal  more.  Suppose, 
for  instance,  that  any  one  said,  '  How  many 
miles  is  it  possible  to  go  up  into  space  V  The 
answer  would  of  course  be,  '  An  infinite  num- 
ber of  miles.'  (Don't  be  frightened  at  this 
continual  occurrence  of  the  word  infinite :  it 
still  means  '  without  any  end,'  and  nothing 
more.)  In  this  case,  if  you  go  a  mile  and  count 
one,  then  another  and  count  two,  and  so  on, 
all  we  mean  is  that  the  process  would  never 
end.  There  would  still  be  space  left  to  go  up 
into,  however  many  millions  of  miles  you  had 
counted.  But  still  all  those  miles  would  be 
counted  and  done  with.  Your  task  would 
have  been  distinctly  begun,  and  there  would  be 
nothing  more  to  say  to  the  miles  behind  you. 


72        THE   CONCEPTION  OF  THE  INFINITE. 

But  try  now  to  count  the  points  in  a  piece  of 
line.  You  count  one,  two,  three,  four,  a  mil- 
lion points ;  and  your  task  is  not  even  begun. 
The  line  is  all  there,  exactly  as  it  was  before  ; 
absolutely  none  of  it  is  done  with.  The  mil- 
lion points  take  up  no  more  line-room  than  one 
point;  that  is  to  say,  absolutely  none  at  all. 
When,  then,  we  are  talking  of  the  points  in  a 
piece  of  line,  we  must  say  not  merely  that  there 
is  a  never-ending  number  of  them  (which  there 
is),  but  that  they  are  out  of  the  reach  of  num- 
ber altogether.  All  the  points  in  a  line  are 
not,  properly  speaking,  a  number  of  points  at 
all.  If  we  are  going  to  speak  about  the  num- 
ber of  points  in  a  line,  we  must  settle  before- 
hand that  we  are  going  to  use  the  word  in  a 
new  sense,  which  is  not  derived  from  counting, 
but  from  this  very  observation  to  which  wc 
have  applied  it. 

"Let  us  now  make  use  of  our  idea  of  a 
path.  When  a  point  moves  along  a  line,  we 
know  that  between  any  two  positions  of  it  there 
is  an  infinite  number  (in  this  new  sense)  of  in- 
termediate positions.  That  is  because  the  mo- 
tion is  continuous.  Each  of  those  positions  is 
where  the  point  was  at  some  instant  or  other. 


KANT,  MILL,  AND  CLIFFORD.  73 

Between  the  two  end  positions  on  the  line,  the 
point  where  the  motion  began  and  the  point 
where  it  stopped,  there  is  no  point  of  the  line 
which  does  not  belong  to  that  series.  We  have 
thus  an  infinite  series  of  successive  positions  of 
a  continuously  moving  point,  and  in  that  series 
are  included  all  the  points  of  a  certain  piece  of 
line-room.  May  we  say,  then,  that  the  line  is 
made  up  of  that  infinite  series  of  points  ? 

"  Yes,  if  we  mean  no  more  than  that  the 
series  makes  up  the  points  of  the  line.  But  no, 
if  we  mean  that  the  line  is  made  up  of  those 
points  in  the  same  way  that  it  is  made  up  of  a 
great  many  very  small  pieces  of  line.  A  point 
is  not  to  be  regarded  as  a  part  of  a  line  in  any 
sense  whatever." 

Evidently  Clifford  saw  more  clearly  than  either 
Kant  or  Mill  that  the  notion  of  quantity  is 
foreign  to  our  conception  of  infinity.  In  speak- 
ing of  the  infinite  extent  of  space,  he  does  not 
make  the  absurd  and  tautological  assertion  that 
it  would  take  an  infinite  time  to  exhaust  it.  He 
claims  that  when  we  call  a  thing  infinite  we 
know  very  well  what  it  means :  it  means  that  it 
will  never  end.  He  never  hints  at  any  possi- 
bility of  knowing  the  infinite  as  a  whole,  recog- 


74        THE   CONCEPTION   OF  THE  INFINITE. 

nizing  very  clearly  that  it  cannot  be  a  whole. 
And  he  has  used  the  word  number  in  two  senses 
(which  a  careful  reading  of  the  extracts  quoted 
will  show  to  be  a  quantitative  and  a  qualitative 
sense),  to  mark  the  difference  between  numbers 
regarded  as  constituent  parts  of  wholes  or  sums 
total,  and  number  regarded  as  unlimited  units, 
having  no  relation  to  wholes  of  any  sort, — a  dis- 
tinction which  precisely  corresponds  to  the  dis- 
tinction which  I  have  drawn  in  a  preceding 
chapter  between  the  quantitative  and  the  quali- 
tative uses  of  the  word  all,  as  applied  to  the 
members  of  a  finite  and  an  infinite  series. 

But  it  is  also  evident  that  Clifford  did  not  see 
the  full  significance  and  value  of  the  truth  which 
he  recognized.  This  is  clear  from  his  contrast- 
ing the  relation  of  linear  miles  to  infinite  exten- 
sion with  the  relation  of  mathematical  points  to 
a  line,  and  indicating  that  in  the  latter  case 
there  is  something  peculiarly  hopeless  in  the  at- 
tempt to  complete  the  line  by  the  addition  of 
such  points.  "  You  count  one,  two,  three,  four, 
a  million  points,  and  your  task  is  not  even 
begun.  The  line  is  all  there,  exactly  as  it  was 
before ;  absolutely  none  of  it  is  done  with."  So 
that  an  addition  of  points,  however  long  con- 


KANT,  MILL,  AND   CLIFFORD.  75 

tinued,  has  no  tendency  to  exhaust  and  complete 
the  line.  But  in  the  case  of  infinite  space,  says 
Clifford,  if  you  go  a  mile  and  count  one,  another 
and  count  two,  and  so  on,  all  those  miles  would 
be  counted  and  done  with.  "  Your  task  would 
have  been  distinctly  begun,  and  there  would  be 
nothing  more  to  say  to  the  miles  behind  you." 
But  in  reality,  so  far  as  they  touch  the  question 
under  discussion,  the  two  cases  are  precisely 
similar.  The  points  have  not  even  begun  to 
exhaust  the  line,  because  their  addition  has  no 
tendency  to  exhaust  it;  their  number  has  noth- 
ing to  do  with  it.*  And,  similarly,  one  may 
justly  hold  that  the  miles  counted  have  not  even 
begun  to  exhaust  the  infinite  space,  since  the 
increasing  of  the  distance  between  limits  has  no 
tendency  to  make  them  approach  the  limits  of 
that  which  is  without  limit ;  since  the  adding  of 
quantities  has  no  tendency  to  make  a  sum  equal  to 
that  which  is  not  a  quantity,  and  cannot  be 
equal  to  anything ;  and  since  adding  mile  to 
mile  has  no  tendency  to  successively  exhaust  the 
parts  of  that  which  is  not  a  whole,  and  can  have 


*   I   speak   here,   of   course,    from   Clifford's    stand-point, 
assuming  the  infinite  divisibility  of  a  line. 


76        THE   CONCEPTION  OF   THE  INFINITE. 

no  parts.  The  relation  between  a  mile  or  a 
thousand  miles  and  infinite  extension  is  no 
closer  than  that  which  Clifford  conceived  be- 
tween a  point  or  a  thousand  points  and  a  given 
line. 

One  cannot  but  see,  therefore,  that  Clifford 
has  not  grasped  the  true  nature  of  our  concep- 
tion of  infinity  in  all  its  consequences  so  thor- 
oughly as  he  might  have  grasped  it.  It  will  be 
seen,  however,  that  he  has  discussed  it  more 
satisfactorily  than  either  of  the  writers  before 
cited. 


CHAPTER    V. 

THE   CONCEIVABLE   AND  THE   EXISTENT. 

There  are  two  quite  distinct  questions  to  be 
considered  in  a  discussion  of  the  infinite :  the 
first  is,  "  Can  an  infinite  object  be  conceived  ?" 
and  the  second,  "  Does  any  infinite  object  really 
exist  V3 

Now,  one  may  answer  the  former  of  these  ques- 
tions in  the  affirmative,  and  yet  not  be  com- 
mitted to  a  similar  answer  to  the  latter.  The 
two  propositions,  "An  infinite  object  is  conceiv- 
able" and  "An  infinite  object  exists"  are  dif- 
ferent in  nature,  and  when  the  former  is  affirmed 
the  latter  still  calls  for  proof.  One  may  hold 
that  space  is  subjective,  a  mere  abstraction  from 
experience  of  extended  objects,  and  that,  conse- 
quently, space,  together  with  the  imaginary  line 
in  space,  exists  only  as  it  is  produced  in  thought; 
and,  though  he  may  on  that  account  deny  that 
that  line,  as  an  imagined  object,  or  any  actually 
existent  line  is  infinite,  or  that  any  line  could 
possibly  be  made  infinite,  he  may  yet  claim  that 

he  can  conceive  an  infinite  line. 

7*  77 


78        THE  CONCEPTION  OF  THE  INFINITE. 

If  he  belong  to  one  school  of  thought  he  will 
not  only  claim  that  he  can  conceive  space  in- 
finite, but  will  assume  on  a  priori  grounds  the 
infinity  of  space  as  actually  existent.  If  he  be  an 
adherent  of  another  school  he  may  hold  that  the 
proposition  "  space  is  infinite"  is  incapable  of 
proof,  and  that  it  can  never  be  maintained ;  but 
he  will  not  on  that  account  deny  that  he  can  con- 
ceive infinite  space.  One  may  maintain  that  our 
assent  to  the  former  proposition  is  conditioned 
on  our  assent  to  the  latter;  that  if  the  infinite 
be  so  unattainable  and  even  contradictory  a  con- 
ception as  Sir  William  Hamilton  has  held,  we 
would  have  no  reason  to  believe  the  existence  of 
any  infinite  object  either  possible  or  actual ;  but 
certainly  no  one  will  hold  that  the  first  of  the 
two  propositions  is  so  based  upon  the  second  as 
to  necessarily  stand  or  fall  with  it.  The  fact 
that  I  can  imagine 

"...  the  Cannibals,  that  each  other  eat, 
The  Anthropophagi,  and  men  whose  heads 
Do  grow  beneath  their  shoulders," 

does  not  prove  such  objects  to  have  real  exist- 
ence. If  men  were  only  able  to  represent  in  the 
imagination  what  has  its  actual  prototype  in 
nature,  how  would  we  account  for 


THE  CONCEIVABLE  AND  THE  EXISTENT.        79 

"  .  .  .  the  pert  Fairies  and  the  dapper  Elves," 

and  the  cloud  of  unreal  creatures  with  which 
the  teeming  poetic  imagination  of  mankind  has 
peopled  the  world  from  earliest  ages  ?  Where 
would  be  the  dragon,  the  basilisk,  the  roc? 
Where  the  valley  of  diamonds  and  the  palace  of 
Aladdin  ?  The  fact  that  I  can  conceive  such  does 
not  prove  that  in  the  whole  realm  of  nature  such 
objects  may  be  found.  And,  similarly,  the  fact 
that  no  objects  of  a  certain  kind  have  an  actual 
existence  is  no  proof  that  the  various  qualities 
representing  them  may  not  be  found  grouped  in 
mind,  as  a  result  of  the  constructive  imagination. 
I  may  examine  in  detail  all  actual  horses  and  all 
actual  men,  and  conclude  that  the  qualities  be- 
longing to  each  animal  are  distinct  and  sepa- 
rate ;  but  I  cannot  assure  myself  that  some  one 
will  not  form  in  his  mind  the  notion  of  an  ani- 
mal which  combines  the  two  sets  of  qualities, — a 
centaur.  All  our  ideas  are  not  derived  in  their 
ultimate  shapes  from  direct  intuition  of  objects, 
but  are  a  result  of  mental  processes,  which  recast 
and  variously  combine  the  material  furnished. 
Though  we  may  not  be  able  to  add  a  new 
element  to  those  simple  factors  of  our  experi- 
ence primarily  given,  yet  it  cannot  be  denied  that 


80       THE   CONCEPTION  OF    THE  INFINITE. 

we  may  so  transpose  or  combine,  add  or  elimi- 
nate, some  of  those  elements,  as  to  obtain  a  prod- 
uct consisting  of  elements  which  are  not  found 
thus  combined  in  any  single  object  as  it  is  pre- 
sented to  us  in  nature.  When  it  is  asked,  there- 
fore, whether  any  given  conception  be  conceiv- 
able, it  is  not  necessary  to  an  affirmative  answer 
that  any  existent  object  be  proved  to  correspond 
to  the  conception.  It  is  enough  if  the  qualities 
connotated  by  the  name  be  shown  to  be  such  as 
may  be  mentally  put  together,  and  are  not  mutu- 
ally repugnant  or  contradictory.  We  find  thus 
that  the  answer  to  the  first  question  put  forward 
at  the  beginning  of  the  chapter  does  not  depend 
on  the  answer  to  the  second,  and  may  be  true 
though  that  be  false.  When  it  is  asked  whether 
the  answer  to  the  second  question  may  not  be 
independent  of  the  answer  to  the  first,  the  an- 
swer is  not  so  clearly  affirmative ;  and  although 
this  point  is  not  directly  connected  with  the  sub- 
ject under  discussion,  I  may  say,  a  proptos  of  thv 
antagonism  which  Hamilton  exhibits  between 
Faith  and  Knowledge,  that,  were  the  things  of 
which  I  have  just  spoken  so  defined  that  I  could 
in  no  conceivable  manner  put  together  in  mind 
the  contradictory  qualities  attributed  to  them,  I 


THE  CONCEIVABLE  AND  THE  EXISTENT.       gl 

should  be  doing  rashly  to  assume  their  ex- 
istence, however  veracious  the  witness  upon 
whose  word  their  acceptance  might  depend.  It 
may  be  a  question  whether  any  force  can  be 
brought  to  bear  upon  a  man  sufficiently  strong 
to  lead  him  to  believe  in  the  actual  existence  of 
an  inconceivable  object;  that  is,  of  that  which 
cannot  become  an  object  of  thought,  nor,  conse- 
quently, of  belief.  However,  the  point  in  which 
we  are  now  chiefly  concerned  is  this :  that  there 
are  a  vast  number  of  things  that  Ave  have  no 
reason  to  believe  in  as  actually  existent,  but 
which  we  nevertheless  conceive  with  perfect 
clearness. 

Again.  When  we  put  forward  the  proposi- 
tion that  an  infinite  object  is  conceivable; 
unless  we  examine  carefully  this  term  used, — 
conceivable, — and  find  out  whether  it  has  one 
meaning  or  two,  we  may  involve  ourselves  in 
serious  error.  If  the  word  has  two  meanings, 
or  is  in  common  use  made  to  stand  for  two 
distinct  mental  operations ;  when  we  prove  that 
an  infinite  object  is  inconceivable  in  one  ac- 
ceptation of  the  term,  we  do  not  necessarily 
prove  that  it  is  inconceivable  in  the  other 
sense.     If  we  suppose  that  in  its  first  meaning 


82       THE   CONCEPTION  OF  THE  INFINITE. 

the  word  conceivable  is  synonymous  with  im- 
aginable, and  that,  when  we  eliminate  one  or 
more  of  the  qualities  which  must  be  present 
to  make  an  object  imaginable,  we  may  still  in 
some  way  mentally  grasp  the  remaining  quali- 
ties, this  second  operation  we  may  call  con- 
ceiving in  the  second  and  narrower  sense  of  the 
term.  The  proof  that  there  is  such  a  mental 
operation  I  will  reserve  for  the  next  chapter. 

Let  it  be  marked,  therefore,  that  we  have  to 
consider  three  distinct  propositions,  which  de- 
mand as  many  distinct  kinds  of  proof,  when 
we  assert,  in  the  first  place,  that  an  infinite  ob- 
ject is  conceivable,  and  in  the  second,  that  an 
infinite  object  exists. 

If  I  assert,  then,  that  I  can  conceive  space 
to  be  infinite,  the  proof  should  consist  in  an 
analysis  of  the  qualities  represented  by  the 
word,  and  an  examination  of  their  mutual  con- 
sistency or  repugnancy.  If  one  element  com- 
prehended in  the  conception  that  I  have  in 
mind  be  the  notion  of  a  certain  quantity,  then 
I  must  in  some  way  mentally  grasp  that  quan- 
tity and  connect  it  with  the  other  elements  in 
the  conception,  as  I  do  in  the  representation 
in  imagination  of  finite  objects.     If  there  be  in 


THE  CONCEIVABLE  AND  THE  EXISTENT       83 

the  conception  no  quantitative  element,  all  that 
is  necessary  is  the  ability  to  grasp  in  thought 
certain  qualitative  elements,  and  this  would  fall 
under  the  second  and  narrower  sense  of  the 
term  conception.  In  either  case  the  proposi- 
tion depends  on  no  fact  of  actual  being,  and 
may  be  established  without  any  reference  to 
such. 

But  the  proposition  "  space  is  infinite"  de- 
mands in  the  way  of  proof  what  the  previous 
propositions  can  dispense  with.  Admitting  my 
ability  to  conceive  infinite  space,  I  may  yet 
doubt  its  existence,*  which  doubt  I  may  pro- 
ceed to  remove  by  proof.  I  may  assume  on 
a  priori  grounds  the  existence  of  infinite  space, 
as  a  postulate  of  Reason;  or  I  may  deny  the 
possibility  of  establishing  the  proposition  a 
priori  and  be  forced  to  fall  back  upon  obser- 
vation, which  proceeds  by  adding  space  to 
space.  In  the  latter  case,  although  I  may  ad- 
mit that  the  conception  of  the  infinite  is  qual- 
itative, and  means  in  the  case  in  point  only 
that  a  progression   along   a  given   line  may  be 

*  I  purposely  avoid  all  consideration  of  metaphysical  argu- 
ments drawn  from  theories  as  to  the  nature  of  space. 


84        THE   CONCEPTION  OF    THE  INFINITE. 

endless,  yet  the  only  apparent  possibility  of  es- 
tablishing this  fact  with  reference  to  any  given 
line  lies  in  a  continued  addition  of  parts.  That 
is,  though  the  conception  itself  may  be  quali- 
tative, and  contain  no  reference  to  a  whole  and 
parts,  the  only  way  in  which  it  seems  possible 
to  prove  any  actual  object  infinite  is  by  em- 
ploying the  ideas  of  quantity  and  totality  in 
adding  part  to  part,  which  ideas  are  in  fact 
contradictory  to  ideas  already  contained  in  the 
conception. 

Now,  it  is  evident  that  observation,  adding 
part  to  part,  could  never  attain  the  infinite, 
for  the  infinite  time  in  which  it  is  said  to  be 
possible  to  accomplish  this  is  no  quantity  of 
time  at  all,  and  the  phrase  merely  expresses  in 
a  disagreeably  tautological  and  roundabout  way 
that  it  can  never  be.  But  in  spite  of  this, 
since  our  experience  in  going  from  part  to 
part  seems  to  be  analogous  to  what  would  be 
our  experience  of  the  object  if  it  were,  indeed, 
infinite;  and  since  there  seems  to  be  no  other 
way  of  proving  from  experience  its  infinity,  we 
are  apt  to  imagine  that  by  this  process  we  are' 
somehow  proving  the  infinity  of  an  object,  or 
are  at  least  in  the  way  to  prove  it.     Evidently 


THE  CONCEIVABLE  AND  THE  EXISTENT.        85 

this  feeling  was  in  the  mind  of  Kant  when  he 
put  forward  the  first  of  his  antinomies ;  and  it 
is  also  evident  that  it  exerted  its  influence  on 
the  mind  of  Hamilton,  leading  him  to  "  rise 
from  sphere  to  sphere  in  the  region  of  empty 
space"  in  the  fruitless  endeavor  to  exhaust  in- 
finite space.  The  same  feeling  it  was  that 
caused  Clifford  to  affirm  that,  when  we  have 
counted  one,  two,  three,  four  miles  up  into 
space,  although  we  must  admit  that  the  attempt 
can  never  give  us  infinite  space,  we  may  yet 
regard  our  task  as  "  distinctly  begun."  The 
attempt  must  fail,  he  thought,  hut  this  is  the 
only  way  even  to  make  the  attempt. 

Now,  if  Sir  William  Hamilton  had  set  before 
himself  the  task  of  proving  that  we  can  never 
know  space  to  be  infinite,  his  argument  would 
have  been  to  the  point.  If  he  denied  the 
possibility  of  knowing  space  to  be  infinite  a 
■priori,  no  method  of  proving  it  such  was  left 
except  the  method  of  observation.  He  could 
not,  of  course,  reasonably  hold  that  the  em- 
pirical method  is  a  true  method  of  proving 
anything  to  be  infinite,  but  he  might  prove 
with  the  arguments  that  he  has  used  for  an- 
other   purpose    that    the    only    even    apparent 

8 


86       THE   CONCEPTION  OF  THE  INFINITE. 

method  of  proving  space  infinite  is  merely  ap- 
parent, and  that  consequently  we  can  never 
know  space  to  be  infinite. 

Or  if  Sir  William  had  set  before  himself  the 
task  of  proving  that  we  cannot  imagine  or  call 
up  before  the  mind  an  exact  and  complete  rep- 
resentation of  an  infinite  line,  his  arguments 
would  not  have  been  aside  from  the  point. 
The  imagination  can  picture  but  a  small  ex- 
tent of  line  at  any  one  time ;  this  portion  must 
consequently  be  limited ;  each  portion  succes- 
sively added  in  imagination  must  also  have  its 
limits;  and  we  have  no  escape  from  the  yery 
difficulty  which  we  have  before  met  in  the  at- 
tempt to  know  an  object  as  infinite, — the  im- 
possibility of  getting  rid  of  the  limits  altogether. 
Whether  we  conceive  our  conception  of  an  in- 
finite line  to  be  quantitative  or  qualitative, 
there  is  an  equal  impossibility  of  getting  rid 
of  the  quantitative  in  trying  empirically  to 
prove  a  line  infinite  and  in  imagining  an  infi- 
nite line. 

Sir  William  accepted  this  argument  as  a  proof 
of  the  inconceivability  of  the  infinite.  Now,  an 
argument  which  will  prove  any  infinite  object 
unknowable  as  an  actually  existing  thing  will 


THE  CONCEIVABLE  AND  THE  EXISTENT.       87 

prove  an  infinite  object  unimaginable  only  on 
the  supposition  that  the  element  which  has 
caused  the  impossibility  in  the  former  case  be 
also  present  in  the  latter.  An  argument  which 
would  prove  that  I  cannot  know  a  line  to  be  in- 
finite, because  I  cannot  in  the  attempt  to  prove 
it  such  get  rid  of  the  element  of  quantity,  would 
prove  that  I  cannot  imagine  an  infinite  line,  pro- 
vided that  the  attempt  to  imagine  such  a  line 
must  be  similar  in  kind  to  the  former  process, 
and  must  meet  the  same  difficulty  with  the 
quantitative  element.  And,  similarly,  an  argu- 
ment which  would  prove  an  infinite  line  un- 
knowable and  unimaginable  on  account  of  the 
difficulty  occasioned  by  this  quantitative  element, 
would  prove  an  infinite  line  inconceivable  (in  the 
narrower  sense  of  the  term)  only  if  the  element 
is  necessarily  present  in  conceiving  (in  the  nar- 
rower sense)  an  infinite  line  that  has  caused  the 
trouble  in  the  former  cases. 

But  if  we  can  prove  that,  when  the  notion  of 
quantity  is  eliminated  from  a  certain  complex  of 
psychic  elements,  the  remaining  elements  can 
be  in  some  way  grasped  in  mind,  and  are  thus 
grasped  in  certain  actual  and  not  infrequent 
mental  operations,  in  this  case  the  proofs,  which 


88        THE   CONCEPTION  OF    THE  INFINITE. 

might  be  suitable  when  applied  to  the  former 
propositions,  can  have  no  force  whatever. 

When  we  come  in  fact  to  make  a  more  com- 
plete analysis  of  the  elements  comprehended  in 
our  conception  of  the  infinite,  we  will  see  that 
the  arguments  which  have  been  so  often  ad- 
vanced to  prove  the  infinite  inconceivable,  and 
regarded  as  so  unanswerable,  are  really  not  to 
the  point  at  all,  but  may  be  classed  under  the 
old  logical  fallacy  of  ignoratio  elenchi.  Sir  Wil- 
liam Hamilton  proves  inconceivable  something 
which  is  not  at  all  supposed  by  the  phrase  "  an 
infinite  line,"  and  which,  in  fact,  contains  an 
element  flatly  contradictory  to  one  of  those  in- 
dicated by  the  phrase. 

In  view  of  the  foregoing,  the  exhibition  of  the 
danger  in  which  we  stand  of  confounding  three 
distinct  propositions  and  their  appropriate  proofs, 
the  two  points  which  I  would  insist  upon  are 
these :  (1)  That  the  conceivability  of  the  infinite, 
in  the  narrower  sense  of  the  term,  is  quite  dis- 
tinct from  a  knowledge  of  its  existence,  or  from 
its  imaginability,  and  it  is  with  the  possibility  of 
the  first  alone  that  we  have  to  do ;  and  (2)  that, 
whatever  be  the  metaphysical  tenets  embraced 
by  one  as  to  the  nature  of  Space  and  Time,  or 


THE  CONCEIVABLE  AND  THE  EXISTENT.       89 

of  those  universal  laws  of  Being  known  by  one 
school  as  Rational  Intuitions  and  by  an  opposing 
school  as  Highest  Generalizations  from  Expe- 
rience,— that  these  tenets,  however  they  may 
lead  to  the  acceptance  or  rejection  of  the  propo- 
sition which  affirms  the  existence  of  an  infi- 
nite, can  yet  not  affect  the  proposition  which 
affirms  its  conceivability. 


8* 


CHAPTER   VI. 

THE   CONCEIVABILITY   OF   THE   INFINITE. 

When  we  analyze  the  mental  state  in  which 
we  have  reference  to  an  infinite — let  us  take, 
for  example,  an  infinite  line — we  find  the  fol- 
lowing elements :  in  the  first  place,  there  are 
present  the  usual  qualities  of  a  line;  for  the 
fact  of  our  conceiving  it  as  without  limits  need 
not  alter  any  of  its  usual  qualities,  any  more 
than  the  fact  of  our  being  unable  to  see  the 
ends  of  a  telegraph  wire  need  force  us  to  deny 
that  it  is  a  wire  of  a  certain  diameter,  mate- 
rial, or  color;  and,  in  the  second  place,  there 
is  present  the  notion  that,  however  far  we  may 
go  in  thought,  we  will  find  a  continuation  of 
the  line.  In  other  words,  there  is  the  notion  of 
unlimited  ])ossibility  of  quantity, — a  notion  which, 
be  it  marked,  is  strictly  qualitative.  That 
there  is  no  quantitative  element  present  has 
been  insisted  upon  in  previous  chapters.  But 
quantity  in  general,  not  this  or  that  quantity, 

is  as  much  a  qualitative  notion  as  color  or  form; 
90 


THE  CON  CEIV ABILITY  OF  THE  INFINITE.    91 

and  in  defining  the  second  element  present  in 
our  conception  of  an  infinite  line,  I  have  used 
the  word  advisedly  to  bring  out  what  is  a  dis- 
tinctive characteristic  of  the  conception.  The 
word  infinite  does  not  denote  a  quantity,  but 
it  has  reference  to  quantity,  and  it  cannot,  in 
accordance  with  its  derivation  and  true  signifi- 
cation, be  rightly  applied  to  what  is  incapable 
of  being  quantitatively  considered.  My  objec- 
tion to  the  usage  of  the  word  infinite,  by  some 
who  recognize  that  the  conception  for  which 
it  stands  is  qualitative,  is  that  they  overlook 
the  distinctive  characteristic  of  this  conception, 
which  marks  it  out  from  other  qualitative  con- 
ceptions,— that  is,  its  necessary  reference  to 
quantity,  though  not  itself  quantitative.  If,  by 
that  process  of  abstraction  which  takes  place 
when  I  compare  objects  similar  in  some  of 
their  qualities,  I  fix  my  attention  upon  the 
other  qualities  of  any  finite  line,  disregarding 
its  length,  and  leaving  out  of  view  for  the  time 
being  its  limits,  my  conception  is  qualitative; 
and  yet  it  is  not  the  conception  of  an  infinite 
line.  In  this  case,  so  far  from  affirming  infi- 
nite length,  I  do  not  think  of  length  at  all. 
But,  in   the   case  of  an  infinite  line,  I  add  to 


92       THE   CONCEPTION  OF  THE  INFINITE. 

the  former  complex  of  qualities  a  new  quality, 
possibility  of  quantity  in  general,  not  this  or 
that  quantity.  When  I  try  to  bring  before  my 
mind  the  notion  of  an  infinite  line,  what  I  am 
distinctly  conscious  of  is  this :  I  represent  in 
imagination  a  line  of  indefinite  length,  and 
then  run  mentally  along  the  line  representing 
additional  line-portions, — a  proceeding  which 
would  of  itself  of  course  give  me  only  the 
finite ;  but  what  makes  my  conception  distinc- 
tively of  the  infinite  is  that,  in  this  progression 
or  notion  of  continued  increase,  I  fix  my  at- 
tention upon  the  progression  itself,  and  elimi- 
nate by  abstraction  the  limits  to  which  such  a 
progression  is  subject.  I  do  not,  be  it  marked, 
merely  fix  my  attention  upon  the  other  quali- 
ties of  a  given  line,  abstracting  from  the  notion 
of  limits ;  but  I  have  in  mind  a  progression,  a 
possibility  of  ever-increasing  quantity,  and  I 
abstract  from  the  limits  of  this  progression. 
The  two  conceptions  are  distinctly  different, 
although  both  are  qualitative,  and  they  should 
not  be  confounded  with  one  another. 

The  question,  therefore,  whether  I  can  con- 
ceive an  infinite  line  is  identical  with  the  ques- 
tion whether  I  can  mentally  grasp  the  various 


THE  CO NCEIV 'ABILITY  OF  THE  INFINITE.    93 

qualities  of  a  line,  and  the  notion  of  a  contin- 
ual increase  of  such  a  line,  without  including 
the  notion  of  limits ;  and  it  will  he  seen  that 
this  question  is  simply  one  of  the  phases  of 
the  broader  question  which  is  concerned  with 
the  possibility  of  the  concept  or  general  notion. 
A  certain  complex  of  qualities  being  necessary 
to  the  existence  of  a  given  object  in  nature,  or 
to  its  subjective  existence  as  represented  in  the 
imagination,  is  there  any  mental  operation  by 
which  we  may  grasp  some  of  these  qualities, 
to  the  exclusion  of  others,  and  convey  to  our 
own  and  other  minds  by  the  use  of  the  word 
which  stands  for  this  new  complex  a  distinct 
meaning? 

There  have  been  held  with  reference  to  this 
problem,  as  is  well  known,  three  opinions :  the 
doctrine  of  the  Realists,  that  general  ideas 
have  corresponding  to  them  a  counterpart  Re- 
ality,— a  doctrine  which  may  be  passed  over  as^ 
now  abandoned,  though  its  effects  make  them- 
selves felt  in  many  directions;  the  doctrine  of 
the  Conceptualists,  that,  although  general  ideas 
cannot  exist  in  Nature,  nor  be  represented  in 
the  Imagination,  yet  they  have  a  true  mental 
existence,  and  are  the  result  of  a  distinct  men- 


94       THE   CONCEPTION   OF  THE  INFINITE. 

tal  operation ;  and  the  doctrine  of  the  Nomi- 
nalists, that  the  only  generality  that  has  a 
separate  existence,  subjective  or  objective,  is 
the  Name,  which  may  be  applied  indifferently 
to  many  similar  objects. 

The  Conceptualist  may  hold  that  it  is  possi- 
ble, unless  the  words  include  repugnant  ele- 
ments, to  conceive  an  infinite  line, — that  is,  to 
grasp  in  mind  a  certain  complex  of  psychic 
elements  which  are  yet  incapable  of  being 
pictured  in  the  imagination  as  an  infinite  line. 
To  think,  in  the  sense  of  to  form  such  a  con- 
cept, is  to  him  something  other  than  to  im- 
agine. What  cannot  be  imagined  may  yet  be 
thought.  The  word  man,  which  we  define  as 
comprehending  the  elements  of  rationality  and 
animality,  he  claims,  does  not  in  the  least  in- 
clude all  those  other  qualities  which  must  be 
combined  with  these  two  before  we  can  picture 
in  the  imagination  or  know  as  existing  any 
given  man.  If  we  select  the  two  qualities  in 
which  all  the  objects  of  a  class  resemble  each 
other,  and  give  to  these  two  a  special  name, 
have  we  not  brought  them  into  consciousness 
in  some  way  in  which  we  have  not  the  other 
qualities  ? 


THE  CONCEIVABILITV  OF  THE  INFINITE.    95 

And  when  we  turn  to  the  Nominalist,  it 
would  not  be  hard  to  show  that,  although  his 
doctrines,  if  taken  in  strictness,  would  deny 
the  possibility  of  the  mental  operation  by  which 
we  arrive  at  the  concept,  and  consequently  of 
the  operation  by  which  we  may  grasp  in  thought 
the  various  elements  implied  in  the  phrase  "  an 
infinite  line,"  yet  one  may  find  in  his  teachings 
by  implication  ample  justification  for  assuming 
its  possible  and  actual  existence.  I  will  take 
some  extracts  from  four  well-known  Nominal- 
ists to  show  how  palpable  is  the  fact  stated, 
and  I  will  first  quote  from  Berkeley,  Locke's 
great  opponent  on  the  subject  of  the  Abstract 
Idea  : 

"Whether  others  have  this  wonderful  fac- 
ulty of  abstracting  their  ideas,  they  best  can 
tell ;  for  myself,  I  find,  indeed,  I  have  indeed 
a  faculty  of  imagining  or  representing  to  my- 
self the  ideas  of  those  particular  things  I  have 
perceived,  and  of  variously  compounding  and 
dividing  them.  I  can  imagine  a  man  with  two 
heads,  or  the  upper  parts  of  a  man  joined  to 
the  body  of  a  horse.  I  can  consider  the  hand, 
the  eye,  the  nose,  each  by  itself  abstracted  or 
separated    from   the    rest   of    the   body.     But, 


96        THE   CONCEPTION   OF  THE  INFINITE. 

then,  whatever  hand  or  eye  I  imagine,  it  must 
have  some  particular  shape  and  color.  Like- 
wise the  idea  of  man  that  I  frame  to  myself 
must  be  either  of  a  white,  or  a  black,  or  a 
tawny,  a  straight,  or  a  crooked,  a  tall,  or  a 
low,  or  a  middle-sized  man.  I  cannot  by  any 
effort  of  thought  conceive  the  abstract  idea 
above  described.  And  it  is  equally  impossible 
for  me  to  form  the  abstract  idea  of  motion 
distinct  from  the  body  moving,  and  which  is 
neither  swift  nor  slow,  curvilinear  nor  recti- 
linear; and  the  like  may  be  said  of  all  other 
abstract  general  ideas  whatsoever.  To  be  plain, 
I  own  myself  able  to  abstract  in  one  sense,  as 
when  I  consider  some  particular  parts  or  quali- 
ties separated  from  others,  with  which,  though 
they  are  united  in  some  object,  yet  it  is  pos- 
sible they  may  really  exist  without  them.  But 
I  deny  that  I  can  abstract  from  one  another, 
or  conceive  separately,  those  qualities  which  it 
is  impossible  should  exist  so  separated,  or  that 
I  can  frame  a  general  notion  by  abstracting 
from  particulars  in  the  manner  aforesaid,  which 
last  are  the  two  proper  acceptations  of  abstrac- 
tion. And  there  is  ground  to  think  most  men 
will  acknowledge  themselves  to  be  in  my  case. 


THE  CONCEIV ABILITY  OF  THE  INFINITE.    97 

The  generality  of  men,  which  are  simple  and 
illiterate,  never  pretend  to  abstract  notions.  It 
is  said  they  are  difficult,  and  not  to  be  attained 
without  pains  and  study;  we  may,  therefore, 
reasonably  conclude  that  if  such  there  be,  they 
are  confined  only  to  the  learned."* 

So  much  for  Berkeley's  position  with  respect 
to  the  abstract  notion.  But  mark  the  conces- 
sions which  he  is  forced  to  make  in  a  later 
section : 

"  But  here  it  will  be  demanded,  how  we  can 
know  any  proposition  to  be  true  of  all  partic- 
ular triangles  except  we  have  first  seen  it 
demonstrated  of  the  abstract  idea  of  a  triangle 
which  equally  agrees  to  all?  For,  because  a 
property  may  be  demonstrated  to  agree  to 
some  one  particular  triangle,  it  will  not  thence 
follow  that  it  equally  belongs  to  any  other 
triangle,  which  in  all  respects  is  not  the  same 
with  it.  For  example,  having  demonstrated 
that  the  three  angles  of  an  isosceles  rectangular 
triangle  are  equal  to  two  right  ones,  I  cannot, 
therefore,  conclude  this  affection  agrees  to   all 

*  "Principles  of  Human  Knowledge."    Introduction,  Sect. 
10.     Works,  ed.  Fraser,  vol.  i.  pp.   141,  142. 


98        THE   CONCEPTION  OF   THE  INFINITE. 

other  triangles  which  have  neither  a  right 
angle  nor  two  equal  sides.  It  seems,  there- 
fore, that  to  he  certain  this  proposition  is  uni- 
versally true,  we  must  either  make  a  particu- 
lar demonstration  for  every  particular  triangle, 
which  is  impossible,  or  once  for  all  demonstrate 
it  of  the  abstract  idea  of  a  triangle,  in  which 
all  the  particulars  do  indifferently  partake,  and 
by  which  they  are  all  equally  represented.  To 
which  I  answer,  that  though  the  idea  I  have 
in  view  whilst  I  make  the  demonstration  be, 
for  instance,  that  of  an  isosceles  rectangular 
triangle  whose  sides  are  of  a  determinate  length, 
I  may  nevertheless  be  certain  it  extends  to  all 
other  rectilinear  triangles,  of  what  sort  or 
bigness  soever.  And  that  because  neither  the 
right  angle,  nor  the  equality,  nor  determinate 
length  of  the  sides  are  at  all  concerned  in  the 
demonstration.  It  is  true  the  diagram  I  have 
in  view  includes  all  these  particulars,  but  then 
there  is  not  the  least  mention  made  of  them 
in  the  proof  of  the  proposition.  It  is  not  said 
the  three  angles  are  equal  to  two  right  ones, 
because  one  of  them  is  a  right  angle,  or  be- 
cause the  sides  comprehending  it  are  of  the 
same  length.     Which  sufficiently  shows  that  the 


THE  CO  NCEIV ABILITY  OF  THE  INFINITE.    99 

right  angle  might  have  been  oblique  and  the 
sides  unequal,  and  for  all  that  the  demonstra- 
tion have  held  good.  And  for  this  reason  it  is 
that  I  conclude  that  to  be  true  of  any  obliquan- 
gular  or  scalenon  which  I  have  demonstrated 
of  a  particular  right-angled  equicrural  triangle, 
and  not  because  I  demonstrated  the  proposi- 
tion of  the  abstract  idea  of  a  triangle.  And 
here  it  must  be  acknowledged  that  a  man  may 
consider  a  figure  merely  as  triangular  without 
attending  to  the  particular  qualities  of  the 
angles  or  relations  of  the  sides.  So  far  he 
may  abstract,  but  this  will  never  prove  that 
he  can  frame  an  abstract,  general,  inconsistent 
idea  of  a  triangle.  In  like  manner  we  may 
consider  Peter  so  far  forth  as  man,  or  so  far 
forth  as  animal,  without  framing  the  fore- 
mentioned  abstract  idea  either  of  man  or  of 
animal,  inasmuch  as  all  that  is  perceived  is 
not  considered."* 

In  the  former  of  the  two  extracts  Berkeley 
has  declared  himself  able  to  abstract  only  so 
far  that  he  can  represent  to  himself  in  im- 
agination what  can  exist  separately  in  nature. 

-  "  Principles."     Introduction,  Section  16. 


100     THE   CONCEPTION  OF  THE  INFINITE. 

He  denies  that  he  can  conceive  separately  those 
qualities  which  it  is  impossible  should  exist 
separately.  But  when  he  supposes  an  objector 
to  ask  how  it  is  possible  for  something  proved 
to  be  true  of  a  particular  triangle,  to  be  known 
to  be  true  of  all  triangles,  he  answers  that  it 
is  seen  that  neither  the  right  angle,  nor  the 
equality,  nor  the  determinate  length  of  the 
sides  are  at  all  concerned  in  the  demonstra- 
tion. In  other  words,  he  admits  that,  so  far 
as  that  demonstration  goes,  we  have  to  do  only 
with  those  elements  in  which  all  triangles  agree. 
And  if  we  can  reason  about  certain  elements  to 
the  exclusion  of  others ;  if  we  can  see  that  cer- 
tain objects  are  alike  in  certain  elements  and 
unlike  in  the  others ;  if  we  can  give  a  name  to 
objects  simply  to  express  the  presence  of  these 
same  elements,  however  the  elements  accom- 
panying them  may  vary,  then  surely  the  ele- 
ments of  the  concept  have  been  before  the 
mind  in  some  way  in  which  the  others  have 
not,  and  have  been  grasped  together. 

Berkeley  frankly  admits  as  much  in  the  con- 
cluding sentences  of  the  latter  extract,  sen- 
tences which  were  added  twenty-four  years 
after  the  first  publication   of  the   essay,  when 


THE  CON  CEIV ABILITY  OF  THE  INFINITE.    101 

mature  reflection,  we  may  suppose,  had  brought 
him  to  see  that  on  his  previous  principles,  strictly 
held,  all  comparison  of  objects  differing  in  any 
of  their  qualities  would  be  impossible.  If  we 
can  consider  a  figure  merely  as  triangular,  with- 
out attending  to  the  particular  qualities  of  the 
angles  or  relations  of  the  sides,  then  we  can 
in  some  sort  divorce  the  elements  included 
under  the  general  word  triangle  from  the  ac- 
companying elements  and  consider  them  sepa- 
rately. In  those  last  few  sentences  Berkeley 
admits  all  that  a  reasonable  Conceptualist  would 
care  to  prove,  and  the  words  "abstract  idea," 
as  there  used,  are  equivalent  to  "  object  of  the 
imagination,"  a  something  which  is  not  implied 
in  the  formation  of  the  abstract  or  general 
notion. 

Every  one,  Nominalist  or  Conceptualist,  must 
acknoAvledge  that  we  can  compare  objects  and 
recognize  them  as  like  or  unlike, — not  merely 
like  or  unlike  as  wholes,  but  in  this  or  that 
element;  like  in  length,  unlike  in  breadth;  like 
in  color,  unlike  in  shape.  Now,  no  one  claims 
that  we  can  call  into  clear  consciousness  the 
element  of  length  alone,  and  picture  it  divorced 
of  breadth  and  color ;    but  when  we  recognize 

9* 


102     THE   CONCEPTION  OF  THE  INFINITE. 

two  objects  as  like  in  length  and  unlike  in 
breadth,  the  elements  must  in  some  way  have 
been  present  in  mind  separately,  so  as  to  be 
recognized  as  length  and  breadth.  If  one  object 
that  what  is  present  in  consciousness  must  ipso 
facto  be  perceived,  and  that  we  cannot  perceive 
length  as  a  factor  by  itself,  nor  recall  in  memory 
any  perception  of  such  a  factor  during  the  act 
of  comparison,  I  answer  that  what  is  in  con- 
sciousness is  by  no  means  necessarily  in  a  clear 
analytic  consciousness,  and  that  we  may  by  a 
process  of  deductive  reasoning  be  sure  that 
certain  elements  are  present  as  factors  in  a 
given  mental  state,  while  we  are  yet  quite  un- 
able to  call  these  elements  into  a  clear  analytic 
consciousness,  separated  from  certain  other 
elements  bound  to  them  by  long  association 
and  habit.  As  an  instance,  I  refer  to  vision. 
That  distance  is  itself  unperceivable  by  sight 
we  must  admit.  That  judgments  of  distance 
are  a  result  of  reasoning  from  an  observed 
constant  connection  of  certain  visual  with  cer- 
tain other  elements,  may  be  satisfactorily  es- 
tablished when  the  above  proposition  is  ad- 
mitted. But  to  call  into  clear  consciousness 
by    itself   the    purely    visual    sensation,   which 


THE  CONCEIV ABILITY  OF  THE  INFINITE.    103 

forms  the  basis  of  the  judgment,  is  altogether 
impossible.  That  it  is  a  factor,  and  an  im- 
portant factor,  in  the  complex  consciousness 
which  we  have  at  the  time,  we  know,  and  yet 
its  presence,  as  a  single  and  distinct  element, 
is  capable  of  being  only  deductively  known. 
Notice  a  further  point  which  is  worthy  of  re- 
mark. If  we  vary  the  purely  visual  element, 
allowing  all  the  other  elements  to  remain  the 
same, — that  is,  if  we  change  the  color  of  the 
object,  but  do  not  change  in  any  respect  the 
form  or  size  of  the  image  on  the  retina, — a 
difference  is  at  once  remarked,  and  the  change 
of  color  recognized.  But  the  difference  is  not 
recognized  as  a  difference  between  two  purely 
visual  sensations  when  the  result  of  the  actual 
comparison  comes  into  clear  consciousness,  but 
as  a  difference  in  one  of  their  elements  between 
two  complexes  or  wholes.  That  is  to  say,  the 
two  visual  sensations  cannot  be  brought  into 
clear  consciousness  and  compared  with  each 
other  alone,  but  only  as  they  are  connected 
with  certain  other  elements  in  complexes  or 
wholes;  it  is  the  presence  of  two  or  more  such 
wholes,  which  we  wish  to  compare,  that  pri- 
marily impels  to  the   narrowing  of  the  atten- 


104     THE   CONCEPTION  OF  THE  INFINITE. 

tion  to  the  single  similar  or  dissimilar  elements. 
This  point  is  specially  worthy  of  remark,  as 
there  is  something  closely  analogous  to  this  in 
the  formation  of  the  concept  in  general,  and 
this  special  case  may  help  to  throw  light  upon 
all  cases  in  which  that  which  cannot  be  im- 
agined is  yet  thought. 

When  I  form  the  concept  of  length  by  com- 
paring two  objects  in  length  and  affirming 
agreement,  and  then  recognizing  as  a  distinct 
element  that  in  which  they  agree,  I  certainly 
do  not  compare  the  objects  simply  as  wholes, 
but  compare  the  lengths;  and  I  must  surely 
have  had  these  elements  in  mind  in  some  way 
in  which  I  had  not  the  other  elements  which 
go  to  make  up  the  object.  Whether  I  can  call 
into  clear  consciousness  the  psychic  elements 
present  during  the  operation  or  not,  it  does 
not  much  matter.  I  evidently  have  specialized, 
selected  some  elements  from  among  others,  and 
compared  length  with  length,  as  element  with 
element.  The  name  which  we  give  to  such 
resemblances  is  the  name  representing  a  gen- 
eral or  abstract  idea.  Whether  the  possibility 
of  thus  comparing  single  elements  may  not  be 
always  conditioned   by  the  presence  of  two  or 


THE  CONCEIVABILITY  OF  THE  INFINITE.    105 

more   objects   or   complexes  in  which   the   ele- 
ments are  present  I  will  consider  later. 

Hume  warmly  applauds  the  position  taken 
by  Berkeley  with  reference  to  the  abstract 
idea,  calling  it  "  one  of  the  greatest  and  most 
valuable  discoveries  that  have  been  made  of 
late  years  in  the  republic  of  letters,"  and  he 
undertakes  to  confirm  it  with  proofs  that  he 
hopes  will  put  it  "  beyond  all  doubt  and  con- 
troversy." For  the  same  purpose  for  which 
I  quoted  the  two  extracts  from  Berkeley,  I 
will  quote  the  last  part  of  the  section  which 
he  devotes  to  the  establishment  of  this  posi- 
tion : 

"  It  is  certain  that  the  mind  would  never 
have  dreamed  of  distinguishing  a  figure  from 
the  body  figured,  as  being  in  reality  neither 
distinguishable,  nor  different,  nor  separable,  did 
it  not  observe  that  even  in  this  simplicity  there 
might  be  contained  many  different  resemblances 
and  relations.  Thus,  when  a  globe  of  white 
marble  is  presented,  we  receive  only  the  im- 
pression of  a  white  color  disposed  in  a  certain 
form,  nor  are  we  able  to  separate  and  distin- 
guish the  color  from  the  form.  But  observing 
afterwards  a  o-lobe  of  black  marble  and  a  cube 


106     THE   CONCEPTION  OF  THE  INFINITE. 

of  white,  and  comparing  them  with  our  former 
object,  we  find  two  separate  resemblances  in 
what  formerly  seemed,  and  really  is,  perfectly 
inseparable.  After  a  little  more  practice  of  this 
kind  we  begin  to  distinguish  the  figure  from 
the  color  by  a  distinction  of  reason, — that  is,  we 
consider  the  figure  and  color  together,  since 
they  are,  in  effect,  the  same  and  undistinguish- 
able,  but  still  view  them  in  different  aspects,  ac- 
cording to  the  resemblances  of  which  they  are 
susceptible.  When  we  would  consider  only  the 
figure  of  the  globe  of  white  marble,  we  form  in 
reality  an  idea  both  of  the  figure  and  color, 
but  tacitly  carry  our  eye  to  its  resemblance 
with  the  globe  of  black  marble;  and  in  the 
same  manner,  when  we  would  consider  its 
color  only,  we  turn  our  view  to  its  resemblance 
with  the  cube  of  white  marble.  By  this  means 
we  accompany  our  ideas  with  a  kind  of  reflec- 
tion, of  which  custom  renders  us,  in  a  great 
measure,  insensible.  A  person  who  desires  us 
to  consider  the  figure  of  a  globe  of  white  mar- 
ble without  thinking  on  its  color,  desires  an 
impossibility;  but  his  meaning  is  that  we 
should  consider  the  color  and  figure  together, 
but   still   keep  in    our   eye  the   resemblance  to 


THE  CON  CEIV ABILITY  OF  THE  INFINITE.    J 07 

the  globe  of  black  marble,  or  that  to  any  other 
globe  of  whatever  color  or  substance."* 

It  is  not  hard  to  see  that  we  cannot  distin- 
guish in  a  body  figured  "  many  different  re- 
semblances and  relations"  without  bringing  the 
resembling  elements  in  some  sense  singly  into 
thought;  if  the  mental  complex  which  we  call 
an  object  were  an  indissoluble  unit,  we  could 
affirm  a  general  likeness  or  unlikeness  between 
it  and  other  objects,  but  we  could  not  affirm 
that  the  resemblance  lay  in  the  figure  or  color. 
If,  as  Hume  asserts,  the  figure  and  color  "  are, 
in  effect,  the  same  and  undistinguishable,"  why 
do  we  find  the  one  susceptible  of  the  one  class 
of  resemblances  and  the  other  of  another  class  ? 
If  we  take  the  words  literally,  should  not  the 
figure,  viewed  in  one  aspect,  be  susceptible  of 
resemblances  of  figure,  and  viewed  in  another 
of  color  ?  And,  similarly,  if  the  color  is  one 
with  the  figure, — the  same  and  undistinguisha- 
ble,— should  not  the  color,  viewed  in  one  aspect, 
be  susceptible  of  resemblances  of  color,  and 
viewed  in  another  of  figure  ?     Hume's  admis- 

*"  Treatise  of  Human  Nature,"  bk.  i.  Sect.  7.  Works, 
Boston,  1854,  vol.  i.  p.  42. 


108     THE  CONCEPTION   OF  THE  INFINITE. 

sion  that  the  two  elements  are  known  as  giving 
different  resemblances,  in  itself  refutes  his  pre- 
vious assertion  that  they  are  undistinguishahle. 
If  color  he  recognized  as  like  color,  and  figure 
like  figure,  the  two  qualities  are  distinguished 
as  different,  and  are  in  reality  separately 
grasped. 

I  will  now  take  a  passage  from  Mr.  J.  S. 
Mill's  "  Examination  of  Sir  William  Hamil- 
ton's Philosophy" : 

"  The  formation,  therefore,  of  a  Concept 
does  not  consist  in  separating  the  attributes 
which  are  said  to  compose  it  from  all  other 
attributes  of  the  same  object,  and  enabling  us 
to  conceive  those  attributes,  disjoined  from 
any  others.  We  neither  conceive  them,  nor 
think  them,  nor  cognize  them  in  any  way  as  a 
thing  apart,  but  solely  as  forming,  in  combina- 
tion with  numerous  other  attributes,  the  idea 
of  an  individual  object.  But,  though  thinking 
them  only  as  part  of  a  larger  agglomeration, 
we  have  the  power  of  fixing  our  attention  on 
them  to  the  neglect  of  the  other  attributes 
with  which  we  think  them  combined.  While 
the  concentration  of  attention  actually  lasts, 
if  it  is  sufficiently  intense,  we  may  be  tempo- 


THE  C0NCE1VAB1L1TY  OF  THE  INFINITE.    109 

rarity  unconscious  of  any  of  the  other  attri- 
butes, and  may  really,  for  a  brief  interval, 
have  nothing  present  to  our  mind  but  the  at- 
tributes constituent  of  the  concept.  In  gen- 
eral, however,  the  attention  is  not  so  com- 
pletely exclusive  as  this;  it  leaves  room  in 
consciousness  for  other  elements  of  the  con- 
crete idea;  though  of  these  the  consciousness 
is  faint  in  proportion  to  the  energy  of  the  con- 
centrative  effort,  and  the  moment  the  attention 
relaxes,  if  the  same  concrete  idea  continues  to 
be  contemplated,  its  other  constituents  come 
out  into  consciousness.  General  concepts,  there- 
fore, we  have,  properly  speaking,  none;  we 
have  only  complex  ideas  of  objects  in  the  con- 
crete ;  but  we  are  able  to  attend  exclusively  to 
certain  parts  of  the  concrete  idea ;  and  by  that 
exclusive  attention  we  enable  those  parts  to  de- 
termine exclusively  the  course  of  our  thoughts 
as  subsequently  called  up  by  association,  and 
are  in  a  condition  to  carry  on  a  train  of  medi- 
tation or  reasoning  relating  to  those  parts  only, 
exactly  as  if  we  were  able  to  conceive  them 
separately  from  the  rest."* 

*  "  Examination  of  Sir  William  Hamilton's  Philosophy," 

vol.  li.  p.  64.     Boston,  1868. 

10 


HO     THE   CONCEPTION  OF  THE  INFINITE. 

This  passage  is  so  clearly  in  harmony  with 
the  views  of  the  Conceptualist,  as  I  have  por- 
trayed them,  that  it  seems  scarcely  necessary 
to  comment  upon  it.  But  I  cannot  resist  the 
temptation  to  delay  for  a  moment  over  an  in- 
consistency into  which  Mill  was  forced  by  his 
attempt  to  recognize,  though  a  Nominalist,  a 
truth  which  the  Nominalist,  pure  and  simple, 
cannot  recognize.  The  formation  of  a  Concept, 
he  insists,  does  not  consist  in  "  separating  the 
attributes  said  to  compose  it  from  all  other 
attributes  of  the  same  object,  and  enabling  us 
to  conceive  those  attributes,  disjoined  from  any 
others."  This  position  lie  emphasizes  by  the 
further  affirmation  that  "  we  neither  conceive 
them,  nor  think  them,  nor  cognize  them  in  any 
way  as  a  thing  apart,  but  solely  as  forming,  in 
combination  with  numerous  other  attributes, 
the  idea  of  an  individual  object."  These  sen- 
tences are  certainly  unequivocal :  they  con- 
tain an  emphatic  assertion  of  the  nominalistic 
doctrine. 

But,  side  by  side  with  such  statements,  we 
find  it  asserted  that  we  may  fix  the  attention 
upon  the  attributes  constituent  of  the  concept, 
to  the  neglect   of  the   other   attributes  of  the 


THE  CO  NCFAV ABILITY  OF  THE  INFINITE.    H\ 

object,  and  that  while  the  concentration  of  at- 
tention actually  lasts,  if  it  is  sufficiently  intense, 
"  we  may  be  temporarily  unconscious  of  any  of 
the  other  attributes,  and  may  really,  for  a  brief 
interval,  have  nothing  present  to  our  mind  but 
the  attributes  constituent  of  the  concept." 
Surely,  if  the  only  elements  before  the  mind 
are  those  constituent  of  the  concept;  if  we 
may  be  conscious  of  these,  even  for  a  brief 
interval,  and  conscious  of  these  alone;  surely 
in  such  a  case  we  conceive,  or  think,  or  in  some 
way  cognize  the  attributes  forming  the  concept 
as  separate  and  apart,  and  not  for  the  time 
being,  in  combination  with  numerous  other 
attributes.  Mr.  Mill  goes  even  further  in  the 
above  admission  than  most  of  us  would  care  to 
follow  him.  In  speaking  as  he  does  of  the 
process,  and  not  distinguishing  between  the 
imagining  of  an  object  and  the  knowing  of  one 
or  more  of  its  isolated  qualities,  he  clearly  in- 
timates, although  he  does  not  distinctly  say, 
that  the  elements  before  the  mind  during  the 
formation  or  use  of  a  concept  are  in  conscious- 
ness in  the  same  way  that  the  whole  complex  or 
object  may  be  in  consciousness.  But,  to  recur 
to  the  before-mentioned  analogy  of   the  purely 


112     THE    CONCEPTION   OF   THE  INFINITE. 

visual  element  in  vision,  Ave  know  that,  although 
we  may  so  concentrate  the  attention  as  to  dis- 
tinguish the  hlue  color  of  one  object  from  the 
red  color  of  another,  and  so  must  have  com- 
pared in  some  rapid  manner  these  purely  visual 
sensations,  yet  when  we  try  to  call  into  clear 
consciousness  the  mere  sensation  of  color,  we 
cannot  do  it  without  imagining  the  color  as 
on  a  surface,  or  combined  with  psychic  ele- 
ments not  purely  visual.  That  is  to  say,  the 
single  and  separate  sensations  cannot  be  called 
into  a  clear  consciousness,  and  their  presence 
when  we  use  the  concept,  or  have  occasion  to 
compare  them  singly  with  each  other,  is  some- 
thing quite  distinct  and  different  from  the  pres- 
ence in  consciousness  of  the  complex  which  is 
knowable  as  an  object.  And  such  would  seem 
also  to  be  the  case  wherever  we  call  before 
the  mind  the  single  psychic  elements  which  can 
yet  not  be  represented  alone  in  the  imagination. 
The  element  must  have  been  grasped  separately, 
but  it  can  be  brought  into  a  clear  consciousness 
only  in  combination. 

If  now  we  recognize  in  each  of  two  objects 
presented  to  us  a  certain  quality  or  complex 
of  qualities  upon  which  we  can  1ix  the  atten- 


THE  CON CFAV ABILITY  OF  THE  INFINITE.    \\% 

tion,  and  if  we  discover  that,  so  far  as  these 
qualities  go,  there  is  an  undistinguishable  sim- 
ilarity in  the  objects,  the  differences  arising  al- 
together from  other  qualities,  why  may  we  not 
call  the  complex  of  qualities  in  point  a  general 
notion  or  general  idea  f  Of  course,  whether  we 
should  call  the  qualities  in  the  two  instances 
the  same,  even  if  they  were  undistinguishably 
similar,  would  depend  on  our  use  of  the  word 
same,*  and  our  ideas  of  what  constitutes  same- 
ness or  identity;  but  I  can  see  no  objection  to 
using  the  words  "  general  notion"  to  indicate 
the  fact  that  a  certain  complex  of  qualities  is 
to  be  found  in  many  different  combinations 
with  other  qualities.  Should  it  still  be  insisted 
that,  since  we  cannot  bring  separately  into 
clear  consciousness  these  elements  of  objects 
known,  we  have  no  reason  to  assume  that 
we  actually  conceive  them  or  think  them  sep- 
arately, I  will  not  quarrel  over  the  use  of 
a  word,  but  will  simply  state  that  I  find  the 
word  "  conceive"  a  useful  one  to  express  that 
concentration    of    the    attention    upon    certain 


*  I    have   pointed    out    before    the    fact    that    the    word 
'same"  is  commonly  used  in  four  quite  distinct  senses. 
10* 


114     THE   CONCEPTION   OF   THE  INFINITE. 

qualities  of  an  object  which  takes  place  when 
objects  are  compared,  and  which  eliminates 
from  consciousness,  or  at  least  subordinates 
all  other  qualities  of  the  objects;  and  I  will  so 
use  the  word,  applying  it  to  an  operation  the 
existence  of  which  Mr.  Mill  has  in  so  many 
words  admitted. 

The  last  author  whom  I  will  quote  is  Mr. 
Bain.  I  will  take  some  passages  from  the 
chapter  on  abstraction  in  his  compendium  of 
psychology  and  ethics,  where  he  supports  the 
Nominalistic  doctrine : 

"We  are  able  to  attend  to  the  points  of 
agreement  of  resembling  things,  and  to  neg- 
lect the  points  of  difference,  as  when  we  think 
of  the  light  of  luminous  bodies,  or  the  round- 
ness of  round  bodies.  This  power  is  named 
Abstraction. 

"  It  is  a  fact  that  wTe  can  direct  our  attention 
or  our  thoughts  to  the  points  of  agreement 
of  bodies  that  agree.  We  can  think  of  the 
light  of  the  heavenly  bodies,  and  make  asser- 
tions, and  draw  inferences  respecting  it.  So 
we  can  think  of  the  roundness  of  spherical 
bodies,  and  discard  the  consideration  of  their 
color  and  size.      In  such  an  object  as  the  full 


THE  CONCEIV 'ABILITY  OF  THE  INFINITE.    \\§ 

moon  we  can  concentrate  our  regards  upon  its 
luminous  character,  wherein  it  agrees  with  one 
class  of  objects,  or  upon  its  figure,  wherein  it 
agrees  with  another  class  of  objects.  We  can 
think  of  the  taste  of  a  strawberry,  either  as 
agreeing  with  other  tastes  or  as  agreeing  with 
pleasures  generally.  .  .  ." 

"  Every  Concrete  thing  falls  into  as  many 
classes  as  it  has  attributes;  to  refer  it  to  one 
of  these  classes,  and  to  think  of  the  corre- 
sponding attribute,  are  one  mental  operation. 

"  When  a  concrete  thing  before  the  view  re- 
calls others  agreeing  in  a  certain  point,  our  at- 
tention is  awake  upon  that  point;  when  the 
moon  recalls  other  luminous  bodies,  we  are 
thinking  of  its  light;  when  it  recalls  other 
round  bodies,  we  are  thinking  of  its  roundness. 
The  two  operations  are  not  different  but  iden- 
tical . 

"  On  this  supposition,  to  abstract,  or  to  think 
of  a  property  in  the  abstract,  is  to  classify 
under  some  one  head.  To  abstract  the  prop- 
erty of  transparency  from  water  is  to  recall, 
at  the  instance  of  water,  window-glass,  crystal, 
air,  &c. ;  to  abstract  its  liquidity  is  to  recall 
milk,  vinegar,  melted  butter,  mercury,  &c. ;  to 


116     THE   CONCEPTION  OF  THE  INFINITE. 

abstract  its  weight  is  to  bring  it  into  compari- 
son with  other  kinds  of  gravitating  matter. 

"  Hence  abstraction  does  not  properly  con- 
sist in  the  mental  separation  of  one  property 
of  a  thing  from  the  other  properties,  as  in 
thinking  of  the  roundness  of  the  moon  apart 
from  its  luminosity  and  apparent  magnitude. 
Such  a  separation  is  impracticable ;  no  one  can 
think  of  a  circle  without  color  and  a  definite 
size.  All  the  purposes  of  the  abstract  idea 
are  served  by  conceiving  a  concrete  thing  in 
company  with  others  resembling  it  in  the  at- 
tribute in  question,  and  by  affirming  nothing 
of  the  one  concrete  but  wThat  is  true  of  all 
those  others." 

...  u  In  abstract  reasoning,  therefore,  we 
are  not  so  much  engaged  with  any  single 
thing  as  with  a  class  of  things.  When  we 
are  discussing  government,  we  commonly  have 
in  view  a  number  of  governments  alternately 
thought  of;  if  we  notice  in  any  one  gov- 
ernment a  certain  feature,  we  run  over  the 
rest  in  our  mind,  to  see  if  the  same  feature  is 
present  in  all.  There  is  no  such  thing  as  an 
idea  of  government  in  the  abstract;  there  is 
only  possible  a  comparison  of  governments  in 


THE  CONCEIVABILITY  OF  THE  INFINITE.    H7 

the  concrete ;  the  abstraction  is  the  likeness  or 
community  of  the  individuals."* 

It  will  be  noticed  that  throughout  this  ex- 
tract Mr.  Bain  does  not  distinguish  between 
the  elements  of  an  act  which  come  out  into  a 
clear  consciousness  and  the  elements  which  do 
not  so  come  out,  but  are  necessary  to  the  possi- 
bility of  the  operation.  When  Mr.  Bain  says, 
for  instance,  that  there  is  no  such  thing  as  the 
idea  of  government  in  the  abstract,  but  that 
we  can  compare  governments  in  the  concrete, 
and  recognize  the  likeness  of  the  individuals, 
it  is  perfectly  true  that  all  that  we  are  clearly 
conscious  of  is  several  individual  objects  and 
a  similarity  between  them ;  but  when  we  come 
to  analyze  this  recognition  of  a  similarity,  it 
will  be  s£en  that  the  elements  which  are  known 
as  similar  are  quite  incapable,  by  themselves, 
of  forming  a  concrete  object,  and  yet  they  are 
distinguished  by  the  mind  from  the  dissimi- 
lar elements;  they  must,  therefore,  have  been 
in  some  sort  grasped  separately,  though  they 
cannot  separately  be  brought  into  a  clear  con- 
sciousness.    Whether,  during  this  rapid  act  of 


Mental  and  Moral  Science,"  London,  1868,  pp.  176-78. 


118     THE   CONCEPTION  OF  THE  INFINITE. 

concentration  and  comparison,  the  other  ele- 
ments which  go  to  form  the  object  actually 
disappear  from  consciousness,  or  are  only  dimly 
perceived,  as  Mr.  Mill  suggests  that  they  are  in 
most  cases,  does  not  affect  the  peculiar  char- 
acter of  the  act.  When  I  compare  in  height 
two  trees  which  I  see  side  by  side  in  the  dis- 
tant landscape  before  me,  I  am  perhaps  con- 
scious of  several  objects  in  their  immediate 
vicinity  in  a  dim  and  indefinite  way,  but  the 
two  objects  compared  are  present  in  con- 
sciousness in  a  manner  very  different,  and  are 
grasped,  so  to  speak,  separately.  And  when  I 
fix  my  attention  upon  the  height  of  the  two 
trees,  finding  them  similar  or  dissimilar  in  this 
one  element,  we  have  every  reason  to  suppose 
that  something  very  analogous  takes  place, 
and  that  this  one  element  is  present  in  con- 
sciousness in  some  way  quite  different  from 
the  others,  and  is  grasped  separately  for  the 
time  being.  Were  it  not  so,  we  could  not 
say  the  trees  are  alike  in  height,  but  different 
in  contour  or  color  of  foliage.  We  are  justi- 
fied in  assuming  that  when  we  recognize  two 
trees  as  like  in  height,  but  not  in  color,  we 
have    compared    height  with    height   and   color 


THE  CON  CEIV ABILITY  OF  THE  INFINITE.   H9 

with  color,  and  not  merely  compared  the  one 
object  as  an  undistinguishable  complex  of  qual- 
ities with  another  object  as  another  undis- 
tinguishable complex. 

As  I  have  before  said,  the  name  which  we 
choose  to  apply  to  this  operation  is  of  little 
consequence;  the  point  chiefly  to  be  borne  in 
mind  is  that  we  have  here  an  operation  differ- 
ing from  ordinary  imagination,  in  that  it  takes 
cognizance  of  certain  psychic  elements  which 
can  yet  not  be  called  into  clear  consciousness 
by  themselves  as  a  mental  picture.  Whether 
the  two  operations  completely  differ  in  their 
ultimate  nature  is  another  question.  When 
the  Conceptualist  asserts  that  though  he  can- 
not imagine  length  apart  from  breadth  or 
color,  yet  he  can  conceive  or  think  it,  he 
merely  marks  by  a  distinct  name  his  recogni- 
tion of  an  operation  different  from  imagina- 
tion, and  which  is  implied  in  all  comparison 
of  objects.  What  may  be  the  peculiar  psychic 
elements  present  in  the  operation  he  does  not 
necessarily  know,  nor  express  when  he  uses  the 
name.* 

*  Kant  seems  to  have  despaired  of  the  possibility  of  ever 
making  this  analysis:    "  Dieser  Schematismus  unseres  Ver- 


120     THE   CONCEPTION  OF   THE  INFINITE. 

Arguing  from  the  analogy  of  the  purely 
visual  element  in  vision,  one  might  conclude 
that  what  is  actually  present  in  consciousness 
in  comparing  lengths,  for  instance,  is  the  dis- 
tinctive element  which  is  present  in  combina- 
tion with  other  elements  (and,  consequently,  in 
a  modified  form)  in  all  our  experience  of  ex- 
tended objects,  but  which,  in  the  act  of  com- 
paring two  objects,  may  be  brought  into  suf- 
ficient prominence  to  be  considered,  for  the 
moment,  alone,  and  alone  compared  with  its 
kind.  When  we  make  the  attempt  to  call  it 
into  clear  consciousness,  the  element  appears  as 
modified  by,  and  in  combination  with,  others ; 
but  it  is  not  improbable  that,  in  the  act  of 
comparison,  it  obtains  in  its  pure  state  suffi- 
cient recognition  to  make  possible  a  comparison 
with  a  similar  element  also  in  its  pure  state. 
However,  whether  we    can    describe  just  what 

standes,  in  Ansehung  der  Erscheinungen  und  ihrer  blossen 
Form,  ist  eine  verborgene  Kunst  in  den  Tiefen  der  mensch- 
lichen  Seele,  deren  wahre  Handgriffe  wir  der  Natur 
schwerlich  jemals  abrathen  und  sie  unverdeckt  vor  Augen 
legen  werden"  ("  Kritik,  von  dem  Schematismus  der 
reinen  Verstandesbegriffe"),  but  he  did  not  doubt  the 
operation. 


THE  CON  CFAV ABILITY  OF  THE  INFINITE.    121 

is  present  during  the  act  or  not,  we  may  be 
sure  that  a  mental  separation  of  two  objects 
into  their  elements  is  necessary  in  order  to 
a  recognition  of  them  as  in  some  points  simi- 
lar and  in  some  dissimilar. 

Hume  has  asserted  that  if  we  knew  but  the 
one  object,  and  had  no  other  objects  with 
which  to  compare  it,  we  would  never  distin- 
guish between  the  several  elements  composing 
it;  and  the  same  thought  is  made  prominent 
by  Mr.  Bain  when  he  states  that  to  refer  an 
object  to  a  class  of  other  objects,  and  to  think 
of  the  corresponding  attribute,  are  but  one 
act.  Mr.  Bain's  statement  is,  of  course,  some- 
what inaccurate,  as  it  confounds  two  very  dif- 
ferent parts  of  the  one  operation ;  but  the  point 
upon  which  both  of  these  writers  insist, — 
namely,  that  a  comparison  of  objects  is  neces- 
sary to  an  analysis  of  the  objects  into  their 
resemblances, — what  should  be  called  their  ul- 
timate elements, — is  worthy  of  attention.  In 
all  probability,  were  it  not  for  a  comparison 
of  objects,  a  constant  experience  of  groups  of 
psychic  elements  containing  likenesses  and  un- 
likenesses,  we  should  never  analyze  the  groups 

aud  make  prominent  single  elements,  separated 

11 


122     THE  CONCEPTION  OF  THE  INFINITE. 

from  their  accompaniments.  And  since  the 
single  elements  do  not  themselves  come  out 
into  a  clear  analytic  consciousness,  it  is  not  easy 
to  see  how  we  could  be  sure  that  they  had  been 
separately  grasped,  if  we  could  not  infer  their 
presence  from  the  possibility  of  comparison  and 
classification  of  objects.  That  this  analysis  im- 
plies the  presence  of  two  objects,  is  necessarily 
a  classification,  after  the  analytic  habit  has  been 
formed,  as  Mr.  Bain  insists,  is  not  so  clear ;  but 
that,  as  a  preliminary  to  the  act  of  concentra- 
tion by  which  we  form  the  concept,  we  call 
into  consciousness  at  least  one  concrete  object, 
I  think,  cannot  be  doubted ;  and  this  fact  might 
easily  mislead  one  into  taking  the  Nominalistic 
position  that  a  recognition  of  particular  objects 
expresses  the  whole  process.  Particulars  must 
be  present  of  course,  if  they  are  to  be  compared; 
but,  when  compared,  they  are  not  compared  as 
wholes. 

In  view  of  the  foregoing,  I  would,  therefore, 
regard  the  fact  as  beyond  all  doubt,  that  there 
are  mental  operations  differing  distinctly  from 
imagination,  in  that  certain  elements,  of  which 
we  have  usually,  as  single  elements,  no  ana- 
lytic consciousness,  but  which  are  merged  with 


THE  CONCEIVABILITY  OF  THE  INFINITE.    123 

others  into  an  indivisible  whole,  are  brought  for 
the  time  being  into  such  prominence  as  to  be 
compared  individually  with  similar  elements,  and 
recognized  as  like  or  unlike.  It  does  not  follow 
that  we  may  have  a  clear  consciousness  of  the 
steps  in  the  rapid  process  in  which  this  compari- 
son takes  place,  or  clearly  recognize  the  nature 
of  the  elements  compared ;  but  from  the  fact  of 
the  comparison,  about  which  there  can  be  no 
doubt,  we  may  be  very  sure  that  the  operation 
in  question  has  taken  place. 

Now,  when  we  return  to  the  particular  con- 
ception which  we  have  been  considering, — that 
of  an  infinite  line, — we  find  it  merely  a  con- 
crete instance  of  this  general  truth,  which  all 
must  either  explicitly  or  implicitly  admit.  As 
I  have  said,  the  elements  constituent  of  this 
conception  are  the  usual  qualitative  attributes 
of  a  line  and  the  notion  of  continued  pro- 
gression, of  unlimited  possibility  of  quantity. 
These  elements  may  be  brought  into  mind,  to 
the  exclusion  of  the  notion  of  limits,  which 
are  yet  present  in  all  imagined  lines  and  in 
all  intuitions  of  lines  in  nature,  by  employing 
the  process  usual  in  forming  a  concept.  When 
I  think  of  an  infinite  line,  I  first  represent  to 


124     THE   CONCEPTION  OF  THE  INFINITE. 

myself  a  line  of  some  indefinite  length,  and 
I  then  run  mentally  along  this  line,  adding 
new  portions, — that  is,  I  successively  think 
several  increasing  lengths.  I  have  now  before 
my  mind  what  Hume  and  Bain  insist  upon 
and  make  so  prominent  in  forming  the  con- 
cept, several  concrete  objects  similar  in  some 
of  their  qualities.  Having  mentally  passed 
over  several  of  these  line  portions,  I  then  fix 
my  attention,  not  upon  that  in  which  they 
differ, — the  quantitative  element, — but  upon 
that  in  which  they  resemble,  the  usual  quali- 
tative attributes  of  a  line,  and  the  notion  of 
increase  or  progression,  which  is  common  to 
all.  This  is  precisely  what  I  do  in  forming 
the  concept  man  or  animal.  The  concrete  ob- 
jects are  compared,  their  differences  eliminated 
by  abstraction,  and  their  likenesses  grasped  to- 
gether under  a  distinctive  name.  Or  I  may 
select  one  of  the  qualities  in  which  objects 
agree,  and  consider  it  alone,  as  when  I  com- 
pare men  of  the  same  age  and  color,  only  as 
to  their  height,  and  pronounce  them  equal  in 
height.  If  this  be  possible,  if,  in  using  the 
word  man,  I  can  distinguish  between  that  in 
which  men  agree  and  that  in  which  they  clisa- 


THE  CONCEIVABILITY  OF  THE  INFINITE.   125 

gree,  and  if  it  be  further  possible  for  me  to  fix 
my  attention  upon  one  of  the  points  in  which 
they  agree,  to  the  exclusion  of  others,  then  it 
is  possible  to  abstract  from  the  particular  quan- 
tities or  amounts  of  several  lines  present  in  im- 
agination, and  think  onlv  of  a  constant  increase 
or  progression.  That  both  the  one  and  the 
other  are  not  only  possible  but  actual  opera- 
tions, is  proved  beyond  possibility  of  doubt  by 
our  constant  comparison  of  objects,  our  use  of 
general  language,  our  frequent  use  of  the  word 
infinite,  to  indicate  what  is  clearly  distinguished, 
readily  defined,  and  conveys  a  distinct  meaning 
to  speaker  and  hearer. 

One  point  I  will  here  remark  upon  before 
passing,  and  that  is  the  distinction  sometimes 
drawn  between  the  abstract  and  the  general 
notion,  a  point  at  which  I  have  a  few  pages 
before  briefly  hinted.  Admitting  that  Hume 
was  right  in  saying  that  had  we  not  had  oc- 
casion to  compare  two  objects  we  should  never 
have  analyzed  either  into  its  elements,  the  ques- 
tion naturally  arises,  whether,  after  we  have 
formed  this  habit  of  analysis  by  comparison,  we 
may  not  by  mere  effort  of  will  fix  the  attention 

upon  one  element  of  a  single  object,  without  any 
11* 


126     THE   CONCEPTION  OF   THE  INFINITE. 

reference  to  its  occurrence  in  another  object? 
To  take  an  example,  can  I  not,  in  imagining  a 
window  or  a  door,  fix  my  attention  upon  its 
length,  without  thinking  of  the  length  of  any- 
thing else,  or  comparing  the  object  with  any 
other  extended  object? 

The  question  cannot  be  answered  off-hand, 
as  by  saying  that  in  recognizing  my  perception 
as  a  perception  of  length — in  using  the  word 
length — I  necessarily  class  the  object  with 
other  long  objects;  for  it  is  at  least  thinkable 
that  I  may  have  so  associated  the  word  with 
this  peculiar  element  as  to  have  it  suggested 
by  the  presence  of  the  element,  and  still  may 
not  be  conscious  of  other  combinations  in 
which  the  element  occurs.  It  is,  I  think, 
highly  probable,  however,  that  when  we  con- 
centrate attention  upon  one  element  of  an 
object,  there  is  a  more  or  less  dim  and  vague 
reference  to  other  objects,  and  that  there  is  a 
rapid  comparison ;  but  this  fact  must  be  proved 
by  an  interrogation  of  consciousness  during  the 
act,  and  upon  this  point  I  will  not  insist.  If 
it  be  allowed  that  the  one  element  may  be 
known  without  reference  to  its  occurrence  in 
two  or  more  objects,  we  have  what  may  justly 


THE  CONCEIV r ABILITY  OF  THE  INFINITE.   127 

be  called  the  abstract  notion,  as  distinguished 
from  the  element  recognized  as  present  in  sev- 
eral combinations,  in  which  latter  case  we  may 
call  it  the  general  notion.  And  even  if  we 
deny  that  the  abstraction  is  possible  except 
there  be  two  or  more  objects  present  in  mind, 
and  a  comparison  of  them,  yet  it  must  be  ac- 
knowledged that  the  prominence  of  these  ob- 
jects in  consciousness  varies  greatly,  and,  ac- 
cordingly, we  may  have  either  the  intension  or 
the  extension  of  the  concept  most  prominently 
before  the  mind.  If  we  are  concerned,  not  so 
much  with  the  combinations  in  which  the  ele- 
ment occurs  as  with  the  element  itself,  we 
may  call  our  notion  abstract;  if  we  have  prom- 
inently in  mind  the  number  of  occurrences,  we 
may  call  it  general.  In  either  case  the  dis- 
tinction between  the  abstract  and  the  general 
notion  is  a  legitimate  one. 

This  point  is  not  one  directly  connected 
with  the  subject  with  which  I  am  concerned, 
— the  conceivability  of  the  infinite, — but  is  one 
too  interesting  to  be  overlooked  in  any  exami- 
nation into  the  nature  of  the  concept;  and, 
indeed,  it  can  scarcely  be  considered  totally 
foreign  to  the  subject  in  hand,  as  the  operation 


128     THE   CONCEPTION  OF  THE  INFINITE. 

of  forming  a  concept,  and  the  act  of  conceiving 
an  infinite,  are  not  different  in  their  nature,  and 
may  be  viewed  in  the  same  aspects.  And  to 
the  objection  which  may  be  made  to  my  class- 
ing the  notion  of  this  or  that  particular  in- 
finite line  with  the  concept  or  general  notion, 
as  I  have  done  throughout  this  chapter,  the 
objector  taking  the  ground  that  the  individual 
or  the  intuition  is  something  quite  different 
and  distinct  from  the  concept, — to  this  objec- 
tion I  answer  that  the  notion  of  any  particular 
infinite  line  is  not  a  complete  intuition,  in  that 
one  of  the  elements  of  the  intuition  is  elimi- 
nated by  abstraction;  and  that  when,  in  the  for- 
mation of  any  concept,  we  fix  the  attention 
upon  certain  elements  of  an  intuition  to  the 
exclusion  of  others,  we  have  in  mind,  so  to 
speak,  a  constituent  part  of  an  intuition  :  the 
fact  that  we  recognize  its  similarity,  or,  if  we 
so  choose  to  use  the  word,  its  sameness  with 
parts  of  other  intuitions,  does  not  alter  the 
individual  character  of  the  elements  which  we 
actually  have  in  mind.  The  operation  of  form- 
ing a  concept  and  the  operation  of  conceiving 
an  infinite  line  are  in  nature  identical. 

It    seems    impossible    that    any    one,    having 


THE  CONCEIVABILITY  OF  THE  INFINITE.    129 

reflected  upon  the  fact  of  his  constantly  grasp- 
ing in  concepts  elements  which  can  yet  not  be 
separately  imagined,  and  having,  after  an  anal- 
ysis of  what  is  in  his  mind  when  he  calls  up 
the  notion  of  an  infinite,  discerned  the  iden- 
tity of  this  latter  operation  with  the  former,  it 
seems  impossible  that  such  an  one  should  hold 
an  infinite  line,  or  infinite  time  or  space  to  be 
inconceivable.  But,  being  loath  to  give  up  his 
former  position,  such  a  man  will  probably  put 
forward  in  a  new  form  an  objection  upon  which 
I  have  already  commented.  We  have  heard 
him  object,  "  If  we  do  not  know  the  infinite 
as  a  whole,  do  we  not  know  only  its  parts, 
which  are  finite?"  And  now  we  will  hear 
him  object  that,  "  Even  if  it  be  true  that  we 
can  grasp  in  thought  the  notion  of  progression, 
and  the  notion  of  a  line  in  general,  this  will 
give  us  no  knowledge  of  an  infinite  line,  but 
will  give  us  only  the  elements  of  an  incomplete 
image,  which  cannot  be  called  distinctly  before 
consciousness,  and  therefore  cannot  be  known 
as  an  object  at  all."  If,  however,  one  feel 
himself  aggrieved  because  he  cannot  represent 
to  himself,  endowed  with  all  the  qualities  neces- 
sary to  an  object  of  the  imagination,  that  which 


130     THE   CONCEPTION  OF  THE  INFINITE. 

he  has  already  defined  as  wanting  some  of 
those  qualities,  he  will  be  unreasonable  enough 
to  think  it  ground  for  complaint  that  he  can- 
not in  thought  make  parallel  lines  meet,  or 
imagine  a  triangle  with  four  sides.  The  word 
infinite  means  devoid  of  limits,  and  it  neces- 
sarily follows  that  an  infinite  line  cannot  be 
known  as  a  quantity,  consequently  not  as  a 
whole.  Every  object  which  is  seen  or  imag- 
ined has  necessarily  limits,  definite  or  indefi- 
nite: an  infinite  line,  as  infinite,  cannot  become 
an  object  of  the  imagination.  But  from  this  it 
by  no  means  follows  that  I  cannot  call  a  par- 
ticular line  infinite,  provided  I  have  some  proof 
of  the  fact  other  than  its  conceivability,  and 
that  I  cannot  know  my  conception  to  be  in 
harmony  with  the  reality.  Suppose  that,  either 
from  testimony  or  by  means  of  some  a  priori 
chain  of  reasoning,  I  have  good  reason  to  be- 
lieve a  given  line  endless,  I  can  conceive  the 
line  as  without  end,  and  I  may  know  my  con- 
ception, although  it  does  not  represent  the 
total  content  of  my  consciousness  when  at  any 
moment  I  gaze  upon  this  or  that  part  of  the 
line,  to  be  a  true  and  real  conception,  and  in 
harmony    with     my    experience    as    I    progres- 


THE  CON  CEIV ABILITY  OF  THE  INFINITE.   131 

sively  pass  over  the  line ;  and  I  may  be  cer- 
tain that,  however  long  my  experience  may 
continue,  it  will  yet  not  prove  incompatible 
with  the  conception  I  have  formed.  In  this 
sense,  and  in  this  sense  alone,  is  any  infinite 
object  conceivable,  and  there  is  no  other  con- 
ceivable way  in  which  we  could  conceive  it. 
An  infinite  object  which  could  be  known  as  a 
whole  is  not  even  an  object  of  thought,  for 
the  elements  indicated  by  the  words  cannot  be 
so  put  together  as  to  express  a  meaning.  But 
the  conception  of  the  infinite,  as  I  have  de- 
fined it,  contains  in  it  nothing  either  contra- 
dictory or  beyond  the  grasp  of  the  human 
mind,  and  is,  indeed,  a  very  common  concep- 
tion, as  is  evidenced  by  use  of  the  word  infi- 
nite in  literature,  ancient  and  modern,  to  say 
nothing  of  the  constant  occurrence  of  the  word 
in  the  debates  of  those  very  philosophers  who 
find  the  conception  such  a  stone  of  stumbling. 
And  that  the  conception  is  a  real  one,  having 
a  real  consonance  with  experience,  those  of  us 
who  hold  to  the  Christian  doctrine  of  Immor- 
tality will  not  be  slow  to  maintain. 


THE    END. 


1 


f 


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